Color-corrected prism systems

ABSTRACT

A method is described for selecting optical materials to use in designing color-corrected optical systems. Various dioptric, catadioptric and anamorphic optical systems are described, which use only two different types of optical materials to obtain precise axial color correction at three, four or five wavelengths (depending upon the particular optical materials), with only very small chromatic aberration occurring at wavelengths between the precisely color-corrected wavelengths. Examples of color-corrected lens systems comprising more than two glasses are also described.

This application is a division of Ser. No. 419,705 filed on Sept. 20, 1982.

TECHNICAL FIELD

This invention relates generally to color-corrected optical systems. More particularly, this invention pertains to the selection of optical materials for designing optical systems that are color-corrected at more than two wavelengths.

BACKGROUND ART

The optical region of the electromagnetic spectrum is generally understood to extend from the far ultraviolet wavelengths (about 0.36 μm) to the near infrared wavelengths (about 1.10 μm). The selection of optical glasses for designing lens systems that are color-corrected at more than two wavelengths in the optical region has been discussed by a number of investigators, including:

(a) N.v.d.W Lessing, J. Opt. Soc. Am. 47, 955 (1957)

(b) N.v.d.W. Lessing, J. Opt. Soc. Am. 48, 269 (1958)

(c) N.v.d.W. Lessing. Appl. Opt. 9, 1665 (1970)

(d) R. E. Stephens, J. Opt. Soc. Am. 49, 398 (1959)

(e) R. E. Stephens, J. Opt. Soc. Am. 50, 1016 (1960)

(f) M. Herzberger, Optica Acta 6, 197 (1959)

(g) M. Herzberger and N. R. McClure, Appl. Opt. 2, 553 (1963)

(h) M. Herzberger, Optik 35, 1 (1972)

(i) H. Drucks, Optik 23, 523 (1966)

(j) H. Schulz, Optik 25, 203 (1967)

(k) H. Schulz, Optik 25, 208 (1967)

(l) H. Pulvermacher, Optik 30, 297 (1969)

(m) H. Pulvermacher, Optik 30, 396 (1970)

(n) M. Herzberger and H. Pulvermacher, Optica Acta 17, 349 (1970)

(o) B. L. Nefedov, Sov. J. Opt. Technol. 40, 46 (1973)

(p) A. B. Agurok, Sov. J. Opt. Technol. 44, 114 (1977)

(q) M. G. Shoyakin, Sov. J. Opt. Technol. 45, 81 (1978)

(r) M. G. Shpyakin, Sov. J. Opt. Technol. 45, 219 (1978)

(s) G. A. Mozharov, Sov. J. Opt. Technol. 44, 146 (1977)

(t) G. A. Mozharov, Sov. J. Opt. Technol. 47, 398 (1980).

An optical "multiplet" is an optical system comprising a number (designated generally herein by k) of refracting elements made of a number (designated generally herein by q) of different optical materials, where each refracting element is made entirely of a particular optical material, with the various refracting elements in combination providing color correction at a number (designated generally herein by n) of wavelengths. The refracting elements could be configured as lens elements, prism elements, or anamorphic elements, depending upon the nature of the optical system. The term "color correction" as used herein with respect to dioptric and catadioptric systems means correction of axial chromatic aberration. Thus, to say that a lens system or a Mangin mirror system is color-corrected at n wavelengths means that n wavelengths are brought to a common focus on the optic axis of the system. For a prism system, "color correction" means correction of chromatic dispersion from the total deviation angle of the system.

Various combinations of optical glasses were discovered in the prior art for designing lens systems that bring more than two wavelengths to a common focus. Previous lens systems, which were color-corrected at more than two wavelengths, generally required three or more different types of optical glasses for making three or more lens elements. There was little success in the prior art in identifying suitable pairs of optical glasses for designing lens doublets (i.e., lens systems comprising only two lens elements) capable of bringing more than two wavelengths to a common focus.

U.S. Pat. No. 3,395,962 to Herzberger, et al. described a photographic objective comprising three lens triplets, with each triplet being individually color-corrected at four wavelengths. M. Herzberger, in reference (f) on the above list, suggested that color correction of an optical lens system at four wavelengths is substantially equivalent to color correction for the entire spectrum bounded by the most widely separated of those four wavelengths.

A lens system comprising only two lens elements, i.e., a doublet, is inherently simpler than a lens system comprising a triplet or higher-number multiplet of lens elements. In terms of fabrication cost and compatibility with constraints imposed by the practical considerations involved in designing optical instruments, a lens doublet that is color-corrected at three or more wavelengths would ordinarily be preferable to a higher-number lens multiplet color-corrected at the same number of wavelengths. A computer-aided search to identify pairs of optical glasses suitable for designing lens doublets that are color-corrected at four wavelengths was made without success by R. R. Willey, Jr., as reported in Appl. Opt. 1, 368 (1962). However, Willey did report that certain pairs of glasses can be used for designing lens doublets color-corrected at three wavelengths.

An article by M. Gaj and J. Nowak in Optik 29, 321 (1969) stated that lens triplets color-corrected at three equivalents could be designed using various pairs of glasses listed in the optical glass catalog published by Glaswerke Schott of Mainz, West Germany. However, two of the pairs of glasses identified by Gag and Nowak were said to require an additional fluorite lens element to produce the desired color correction at three wavelengths. Gaj and Nowak did not report any design data necessary for verifying the design and for constructing lens elements made from the optical materials identified in their article.

U.S. Pat. No. 3,674,330 disclosed particular pairs of alkali halide crystals from which lens doublets that are substantially achromatic over a broad spectrum of wavelengths in the far infrared region, i.e., at wavelengths longer than 1.5 μm can be designed. However, there have been no reports in the literature of any crystal pairs that can be used for designing lens systems color-corrected at more than two wavelengths in the optical region.

The 1977 article by A. B. Agurok, which is reference (p) on the above list, reported on an investigation of the glasses listed in the Russian optical glass catalog GOST (All Union State Standard) to identify pairs of optical glasses suitable for designing lens doublets color-corrected at more than three wavelengths. The search indicated than no two glass in the GOST catalog can be combined to make a lens doublet that is color-corrected at four wavelengths. Agurok also reported that no two glasses in the Schott catalog had been found, which in combination make a lens doublet that is color-corrected at four wavelengths.

The various algorithms that were developed in the prior art for selecting optical glasses for designing color-corrected lens systems made use of conventional glass parameters (i.e., Abbe numbers and relative partial dispersion ratios) to characterize the dispersive properties of the available types of optical glasses. The approaches taken in the prior art, however, never led to an algorithm of general validity for identifying compatible combinations of optical materials for designing lens multiplets that bring more than two wavelengths to a common focus.

Until the present invention, the problem of correcting optical systems for chromatic aberration has been one of the most difficult problems facing optical designers. A theoretically rigorous procedure for selecting compatible pairs of optical glasses for designing lens doublets that are color-corrected at more than two wavelengths has eluded previous researchers. Non-rigorous "cut-and-try" methods have failed to identify compatible pairs of optical glasses for designing lens doublets color-corrected at more than three wavelengths. C. G. Wynne, in an article entitled "Secondary Spectrum Correction with Normal Glasses", Optics Communications 21, 419 (1977), noted: "It has been generally accepted that the correction of secondary spectrum aberrations, in optical imaging systems, necessarily requires the use of glasses having abnormal relative partial dispersions. This is . . . an error, arising from defects in the accepted theory of first order chromatic aberrations."

The inability of the prior art to develop a unified theory for selecting optical materials to use in designing optical systems that are color-corrected at more than two wavelengths can be attributed to the unavailability, until now, of a tractable mathematical model for representing optical materials in terms of their dispersion characteristics.

DISCLOSURE OF THE INVENTION Buchdahl's Dispersion Equation

The index of refraction (also called the refractive index) of an isotropic homogeneous optical material such as a glass, a crystal or a plastic is a function of the wavelength λ of the optical energy passing through the material. Thus, when an optical material is cut and polished to make a lens element, each wavelength (i.e., each color) is brought to a focus at a different focal position.

Manufacturers of optical glasses conventionally characterize each type of glass in terms of the refractive index of the glass at certain specified wavelengths, viz., the wavelengths of the solar emission lines. In the catalogs published by the leading glass manufacturers, measured values of the refractive index of each glass are listed for the specified wavelengths. In optical design work, however, it is often necessary to know the value of the refractive index of a glass at some wavelength other than those wavelengths specified in the manufactuer's catalog. The refractive index of a glass at any particular wavelength is determined by interpolation from the values given in the catalog by using a so-called dispersion formula.

A great variety of dispersion formulae have been developed over the years to express refractive index as a function of wavelength. A collection of well-known dispersion formulae to given by W. J. Smith in Modern Optical Engineering, McGraw-Hill Book Company, Inc., New York (1966), pp. 147-148. In recent years, a power series dispersion formula derived from a quantum model has come into practically universal use among optical glass manufacturers throughout the world. This power series dispersion formula, which expresses refractive index N in terms of wavelength λ measured in micrometers, is:

    N.sup.2 =A.sub.0 +A.sub.1 λ.sup.2 +A.sub.2 λ.sup.-2 +A.sub.3 λ.sup.-4 +A.sub.4 λ.sup.-6 +A.sub.5 λ.sup.-8, (1)

where the so-called dispersion coefficients A_(j) (j=0, 1, 2, 3, 4, 5), which are provided for each glass by the manufacturer, are obtained by a method of numerical analysis based upon a least squares fit for measured values of the refractive index averaged over several glass melts. All major optical glass manufacturers provide these dispersion coefficients A_(j) for each glass, so that optical designers can compute the refractive index of any optical glass for any desired wavelength.

A formidable obstacle to the development of a tractable mathematical model for representing any optical material in terms of its dispersion characteristics has been the lack of a dispersion formula in the form N=Σa_(n) x^(n) for predicting the refractive index N, where the coefficients a_(n) are characteristic of the particular optical material and are independent of the wavelength λ, and where the variable x is a function of the wavelength, i.e., x=x(λ). None of the dispersion formulae collected by Smith, op. cit., expresses refractive index in this mathematically suitable Σa_(n) x^(n) form; and neither does the quantum model dispersion formula of equation (1).

An expression for refractive index, which provided the starting point for a mathematical model that made the present invention possible, was first introduced by H. A. Buchdahl in a monograph published by Oxford University Press in 1954. This monograph was subsequently reprinted as Optical Aberration Coefficients, Dover Publications, Inc., New York (1968). Buchdahl's analysis began with an expression for refractive index in the form N=N*+c/(λ-λ*), which is known as the Hartmann dispersion formula, where N*, c and λ* are constants that are characteristic of the particular optical material. Buchdahl introduced a change of variable from wavelength λ to a so-called color coordinate, or chromatic coordinate, ω, which allowed the refractive index to be expressed in the form

    N=N.sub.0 +ν.sub.1 ω+ν.sub.2 ω.sup.2 +ν.sub.3 ω.sup.3 + . . . +ν.sub.n ω.sup.n,          (2)

where N₀ is the value of the refractive index at a selected base wavelength λ₀, and where the color coordinate ω is defined as

    ω=δλ/(1+αδλ).        (3)

The quantity δλ is defined as the change in wavelength relative to the base wavelength, i.e.,

    δλ=λ-λ.sub.0,                   (4)

and the term α is a "universal" constant that is substantially independent of the dispersion characteristics of the particular optical material. The coefficients ν_(n), called the dispersion coefficients, are constants characteristic of the particular optical material, and in general vary from one type of optical material to another. P. J. Sands in J. Opt. Soc. Am. 61, 777 (1971), in which he developed the theory of homogeneous lenses, discussed the accuracy of Buchdahl's dispersion formula as given in equation (2).

The color coordinate ω introduced in equation (2) is a function of wavelength λ, and is "universal" in the sense that it does not involve characteristics of the particular optical material. Upon computing values of α for six common optical glasses, Buchdahl found that the values of α a lie within rather narrow bounds for wavelengths in the visible spectrum when the base wavelength λ₀ is chosen to be that of the sodium-D line, i.e, 0.5893 μm. This suggested to Buchdahl that α could be regarded as a "universal" constant for all glasses for wavelengths in the visible spectrum. The value of α calculated by Buchdahl was α=2.5.

Ordinarily, the optical materials used for designing lens systems are optical glasses. For this reason, the analysis presented hereinafter usually refers to optical glasses. However, the analysis is valid for optical materials generally, and is, therefore, applicable regardless of whether the particular optical material under consideration is a glass or some other material such as, e.g., a crystal (e.g., fluorite or quartz), or a viscous liquid (e.g., immersion oil).

The change in refractive index δN as a function of change in wavelength δλ relative to the selected base wavelength λ₀, i.e., δN(λ)=N(λ)-N₀, is defined as the dispersion of the optical material at the wavelength λ relative to the base wavelength λ₀. From equation (2), the dispersion δN of an optical material can be expressed by an equation in the form ##EQU1## which will be referred to hereinafter as Buchdahl's dispersion equation.

A detailed analysis of Buchdahl's dispersion equation (5) for all the various types of optical glasses listed in the latest catalogs published by five leading optical glass manufacturers was undertaken by P. N. Robb and R. I. Mercado, applicants herein. The optical glass manufacturers whose catalogs were used were Schott Optical Glass, Inc., Duryea, Pa.; Ohara Optical Glass Mfg. Co., Ltd., Sagamihara, Japan; Hoya Corporation, Tokyo, Japan; Corning-France, Avon, France; and Chance Pilkington Ltd., St. Asaph, Clwyd, U.K. A total of 813 different types of optical glasses were included in the analysis. An announcement of this analysis was published in an abstract in J. Opt. Soc. Am. 71 (12), 1638 (1981).

The power series in equation (5) converges very rapidly. The convergence was found to be so rapid that all 813 glasses studied could be mathematically modelled using only the first three terms of the power series, with a resulting maximum absolute error of only 0.0001 in the visible spectral range from 0.4 μm to 0.7 μm. It is noted that precision glass melts, on the other hand, can be made to a tolerance of only ±0.0002. Thus, a quadratic dispersion model based upon Buchdahl's dispersion equation, i.e., a dispersion model wherein the degree of the polynomial in equation (5) is given by n=2, can predict the refractive index of an optical glass twice as accurately as the glass can be made.

It was also found that a cubic dispersion model based upon Buchdahl's dispersion equation, i.e., a dispersion model wherein the degree of the polynomial in equation (5) is given by n=3, can predict the refractive index of an optical glass over an extended spectral range from 0.355 μm to 1.014 μm with an accuracy that is approximately equal to the accuracy that can be maintained in making precision glass melts.

It is remarkable that the value of α, for which Buchdahl adopted the "universal" value of 2.5 on the basis of an analysis of only six glasses, was found to vary within the narrow range from 2.299 to 2.960, with an average value of 2.577, for a population of 813 glasses. Buchdahl's value of α=2.5 used in defining the color coordinate ω in equation (3) was therefore virtually exact. Buchdahl calculated the dispersion coefficients ν_(i) of equation (5) using refractive indices for n wavelengths in an nth-degree polynomial and using a base wavelength of λ₀ =0.5893 μm, which is the wavelength of the yellow sodium-D line. Since equation (5) is linear in the dispersion coefficients ν_(i), Buchdahl's procedure yields a system of n simultaneous linear equations in n unknowns. Buchdahl's procedure can be readily implemented with the aid of a computer.

Values for the dispersion coefficients ν_(i) obtained using Buchdahl's procedure were compared with a least-squares solution using seven wavelengths in the visible range (from 0.4 μm to 0.7 μm) and ten wavelengths in the extended spectral range (from 0.365 μm in the ultraviolet to 1.014 μm in the near infrared). It was found that the maximum error could be reduced by a factor of three or more using the least squares method. Adjustment of the least-squares weights and selection of a base wavelength so as to reduce the maximum error further gave a total error reduction factor of five.

R. I. Mercado and P. N. Robb announced in an abstract published in J. Opt. Soc. Am. 71 (12), 1639 (1981) that they had developed a method for selecting optical glasses for two-element lens systems whereby paraxial color correction could be obtained for three, four and five wavelengths. The glass selection method was indicated to be based on use of Buchdahl's color coordinate ω rather than on the use of Abbe numbers and relative partial dispersion ratios. Meanwhile, the method has been generalized to enable selection of glasses for designing multiplet optical systems comprising two or more different types of glasses.

Theoretical Basis for the Invention

In the mathematical model leading to the present invention, the dispersive properties of a particular type of optical glass are described in terms of the dispersive power D of the glass. The dispersive power D is a function of wavelength λ according to the relationship ##EQU2## where N₀ is the value of the refractive index N of the glass at the base wavelength λ₀. The base wavelength λ_(O) may be arbitrarily selected, but is preferably the wavelength at which monochromatic aberrations are calculated for the optical system that is to be designed using the glass.

In developing the theoretical basis for the present invention, the dispersion term δN in equation (6) is replaced by an nth-degree polynomial in the color coordinate ω in accordance with Buchdahl's dispersion equation (5). Thus, equation (6) is rewritten as ##EQU3## where the coefficients η_(i), called the η-dispersion coefficients, are given by η_(i) =ν_(i) /(N₀ -1). Like the ν-dispersion coefficients of equation (5), the η-dispersion coefficients of equation (7) are constants of the particular type of glass and in general vary from one type of glass to another.

The difference in the dispersive power of a particular type of optical glass over a wavelength range λ₁ <λ<λ₂ is represented by D(λ₁,λ₂)=D(λ₁)-D(λ₂). Using equation (7), D(λ₁,λ₂) can be expressed in terms of the color coordinate ω by the power series ##EQU4## For the special case where λ₁ is the hydrogen F line (0.48613 μm), λ₂ is the hydrogen C line (0.65627 μm) and λ₀ is the helium d line (0.58756 μm), it is of interest to note that equation (8) becomes a power series expansion in terms of the reciprocal of the Abbe number for the glass.

Thin Lens Equations

Equations (6), (7) and (8) lead to an elegant expression for the total dioptric power of a lens system comprising a number of thin lens elements in contact. For a single thin lens in air, the dioptric power φ is a function of wavelength λ according to the relationship

    φ(λ)=[N(λ)-1](C.sub.1 -C.sub.2),         (9)

where C₁ and C₂ are the two curvatures of the lens. The curvatures C₁ and C₂ are independent of the wavelength λ, and are conveniently represented by a constant K=C₁ -C₂ that is characteristic of the configuration of the particular lens. Thus, φ(λ)=[N(λ)-1]K. For the base wavelength λ₀, the thin lens power is φ(λ₀)=[N₀ -1]K. Since N(λ)=N₀ +δN(λ), the thin lens power can be written as

    φ(λ)=φ(λ.sub.0)+KδN(λ). (10)

In terms of the dispersive power D as defined by equation (6), equation (10) can be written in the form

    φ(λ)=φ(λ.sub.0)[1+D(λ)].      (11)

In terms of the color coordinate ω, equation (11) can be expressed as ##EQU5## Since ω→0 as λ→λ₀, it follows that φ(λ)→φ(λ₀) as λ→λ₀.

More generally, the total dioptric power Φ of an optical system comprising a number of thin lens elements in contact in air is equal to the sum of the dioptric powers of the individual thin lens elements comprising the system. Thus, for an optical system comprising k thin lens elements in contact having individual lens powers φ₁, φ₂, . . . , φ_(k) at the base wavelength λ₀, the total dioptric power Φ of the system at the base wavelength λ₀ is ##EQU6## The total dioptric power Φ for any wavelength λ is given by ##EQU7## where φ_(j) is the dioptric power of the jth lens element in the system at λ₀, and D_(j) (λ) is the dispersive power of the particular type of glass from which the jth lens element is made. For any wavelength λ, ##EQU8## so that the difference in the total dispersive power of the system over the wavelength range λ₁ <λ<λ₂, i.e., D_(j) (λ₁,λ₂)=D_(j) (λ₁)-D_(j) (λ₂), is given by ##EQU9## and the difference in the total dioptric power of the system at the wavelengths λ₁ and λ₂ is ##EQU10##

Thin Lens Doublet

Applying the general equations (13) to (17) for thin lenses in contact to the particular case of a thin lens doublet (i.e., a lens system for which k=2), the total dioptric power of the doublet at λ₀ is

    Φ(λ.sub.0)=φ.sub.1 +φ.sub.2.            (18)

From equation (14), the dioptric power of the lens doublet for any wavelength λ is

    Φ(λ)=Φ(λ.sub.0)+φ.sub.1 D.sub.1 (λ)+φ.sub.2 D.sub.2 (λ).                (19)

In the case of two separated thin lenses in air, the total dioptric power at λ₀ is

    Φ(λ.sub.0)=φ.sub.1 +(1-k)φ.sub.2,       (20)

where k=dφ₁, where d is the axial separation of the lenses. The parameter k is a function of wavelength according to the relationship k(λ)=k[1+D₁ (λ)].

In the case of a lens doublet comprising two different types of glasses, where the two lens elements of the doublet are cemented together, the condition that the doublet be color-corrected at two wavelengths λ₁ and λ₂ is that Φ(λ₁)=Φ(λ₂). This implies that

    φ.sub.1 D.sub.1 (λ.sub.1,λ.sub.2)+φ.sub.2 D.sub.2 (λ.sub.1,λ.sub.2)=0,                        (21)

where D_(j) (λ₁,λ₂) for j=1, 2 is given by equation (16). Solving the system of equations (18) and (21) yields the dioptric powers of the individual lens elements of the thin lens doublet, to wit:

    φ.sub.1 =-Φ(λ.sub.0)D.sub.2 (λ.sub.1,λ.sub.2)/[D.sub.1 (λ.sub.1,λ.sub.2)-D.sub.2 (λ.sub.1,λ.sub.2)](22)

and

    φ.sub.2 =+Φ(λ.sub.0)D.sub.1 (λ.sub.1,λ.sub.2)/[D.sub.1 (λ.sub.1,λ.sub.2)-D.sub.2 (λ.sub.1,λ.sub.2)]. (23)

The set of equations (22) and (23) is equivalent to the set of equations commonly used for calculating the dioptric powers of the individual lens elements of a two-glass lens doublet color-corrected at the two wavelengths λ₁ and λ₂, where the dioptric powers of the lens elements comprising the doublet are expressed in terms of the Abbe numbers for the glasses from which the lens elements are made.

In the neighborhood of the base wavelength λ₀, the dispersion δN(λ) of a two-glass lens doublet can be approximated by a linear function in the color coordinate ω. Thus, for a two-glass thin lens doublet, equation (5) can be written in the form

    δN(λ)=N(λ.sub.0)+ν.sub.1 ω(λ), (24)

and equations (7) and (8), respectively, become

    D(λ)=η.sub.1 ω(λ)                  (25)

and

    D(λ.sub.1,λ.sub.2)=η.sub.1 [ω(λ.sub.1)-ω(λ.sub.2)].        (26)

Equations (25) and (26) are, respectively, linear models of the dispersive power D(λ) and of the difference in the dispersive power D(λ₁,λ₂) for the two-glass thin lens doublet over the wavelength range λ₁ <λ<λ₂.

Two-Glass Lens Doublet Color-Corrected at Two Wavelengths

In designing a two-glass thin lens doublet that is corrected for paraxial chromatic aberration at two wavelengths, (also called a two-glass two-color lens doublet), it is assumed that the dispersion properties of the optical glasses from which the lens elements of the doublet are made are substantially approximated by equations (25) and (26), i.e., by a linear dispersion model. Thus, each type of optical glass is uniquely represented by two parameters, viz., the refractive index N(λ₀) at the base wave-length λ₀, and the dispersion coefficient η₁ as defined by equation (26). When λ₀ corresponds to the helium d line, and λ₁ and λ₂ are the hydrogen F and C lines, respectively, equation (26) becomes

    [N(λ.sub.F)-N(λ.sub.C)]/[N(λ.sub.d)-1]=η.sub.1 [ω(λ.sub.F)-ω(λ.sub.C)].        (27)

The left side of equation (27) is the reciprocal of the Abbe number. Since the factor [ω(λ_(F))-ω(λ_(C))] is a constant, equation (27) implies that the dispersion coefficient η₁ of an optical glass is directly proportional to the reciprocal of the Abbe number of the glass. The [N(λ₀), η₁ ] glass parameter representation, as suggested by the foregoing analysis, is analogous to the (N_(d), V_(d)) representation commonly used by optical glass manufacturers for uniquely representing an optical glass, where N_(d) and V_(d) are the refractive index and the Abbe number, respectively, for the glass at the wavelength of the helium d line.

In terms of the dispersion coefficients η₁ of the glasses from which the lens elements of a two-glass two-color lens doublet are made, the dioptric powers of the individual lens elements at the base wavelength λ₀ can be expressed by equations in the form

    φ.sub.1 =-η.sub.12 Φ(λ.sub.0)/(η.sub.11 -η.sub.12)                                            (28)

and

    φ.sub.2 =+η.sub.11 Φ(λ.sub.0)/(η.sub.11 -η.sub.12),                                           (29)

where η₁₁ is called the primary dispersion coefficient for the first glass of the doublet, and η₁₂ is called the primary dispersion coefficient for the second glass of the doublet. Ordinarily, it is desirable that the dioptric powers φ₁ and φ₂ of the lens elements of the doublet be small in value, so that the curvatures of the lens elements will be small. From equations (28) and (29), it is apparent that the absolute values of the dioptric powers φ₁ and φ₂ are small when the absolute value of the factor (η₁₁ -η₁₂) is large. Therefore, to obtain a good dioptric power distribution for a lens doublet, there must be a relatively large difference between the primary dispersion coefficients of the glasses from which the lens elements of the doublet are made.

The foregoing analysis has been developed with reference to the η-dispersion coefficients. However, inasmuch as ν₁ is directly proportional to η₁, the analysis is also valid with reference to the ν-dispersion coefficients. The technique of the prior art for designing a two-glass lens doublet that is color-corrected at two wavelengths is thus comprehended as a special case of the algorithm developed for the present invention, wherein a linear approximation of the power series in Buchdahl's dispersion equation (5) is used.

Two-Glass Lens Doublet Color-Corrected at Three Wavelengths

An extension of the foregoing analysis enables selection of compatible glasses for designing a two-glass thin lens doublet that is corrected for axial chromatic aberration at three wavelengths, i.e., a so-called two-glass three-color doublet. To initiate the design of a two-glass thin lens doublet that is color-corrected at wavelengths λ₁, λ₂ and λ₃, where λ₁ <λ₂ <λ₃, the focal length of the doublet may be set equal to unity at the base wavelength λ₀, so that Φ(λ₀)=1 in equation (18). The conditions that must be satisfied to obtain axial color correction of the lens doublet at λ₁, λ₂ and λ₃ imply that Φ(λ₁)-Φ(λ₂)=0 and Φ(λ₂)-Φ(λ₃)=0. These conditions lead to a set of homogeneous equations that are linear in φ₁ and φ₂. Cast in matrix form, these equations can be expressed as

    DΦ=0,                                                  (30)

where ##EQU11##

To express equation (30) in terms of the dispersion coefficients of the optical glasses from which the lens elements comprising the doublet are to be made, a quadratic approximation of the power series in Buchdahl's dispersion equation (5) is used to obtain axial color correction at three wavelengths. More generally, the power series in Buchdahl's dispersion equation (5) is approximated by an (n-1) th degree polynomial in representing a two-glass lens doublet that is color-corrected at n wavelengths.

Assuming that the dispersion properties of each glass of a two-glass three-color lens doublet (i.e., a two-glass three-color lens system comprising k lens elements, where k=1, 2) are approximated with substantial accuracy by a quadratic dispersion model, equation (7) becomes

    D.sub.k (λ)=η.sub.1k ω(λ)+η.sub.2k ω.sup.2 (λ),                                 (31)

and equation (8) becomes

    D.sub.k (λ.sub.i,λ.sub.j)=η.sub.1k (ω.sub.i -ω.sub.j)+η.sub.2k (ω.sub.i.sup.2 -ω.sub.j.sup.2). (32)

In designing a two-glass three-color lens doublet, each type of glass used in making the lens elements of the doublet is uniquely represented by the three parameters N₀ (λ), η₁ and η₂, which comprise the [N₀ (λ), η₁, η₂ ] glass parameter representation. Since η₁ and η₂ are proportional to ν₁ and ν₂, respectively, a [N₀ (λ), ν₁, ν₂ ] glass parameter representation in terms of the ν-dispersion coefficients could also be used for uniquely representing each type of optical glass.

Using the [N₀ (λ), η₁, η₂ ] glass parameter representation relative to the base wavelength λ₀, and specifying three wavelengths λ₁, λ₂ and λ₃ for which color correction is desired, a system of two nonhomogeneous equations can be obtained from equation (32) for each of the k glasses of the lens doublet, where k=1, 2. This system of equations, however, is linear in the dispersion coefficients η_(1k) and η_(2k). In matrix form, this system of equations may be written as

    Ωη.sub.k =D.sub.k,                               (33)

where ##EQU12## Substituting equation (33) into equation (30) yields

    ΩηΦ=0,                                       (34)

where ##EQU13## The elements of each column vector of the matrix η are the dispersion coefficients of the glasses from which the lens elements of the two-glass three-color doublet are made. The matrix element η₁₁ is called the primary dispersion coefficient of the first glass of the doublet, and the matrix element η₂₁ is called the secondary dispersion coefficient of the first glass of the doublet. Similarly, the matrix elements η₁₂ and η₂₂ are called the primary and secondary dispersion coefficients, respectively, of the second glass of the doublet.

The matrix Ω in equation (33) may be factored in the following form ##EQU14## Since Ω is nonsingular, it has a unique inverse. Hence, if equation (34) is multiplied by the inverse of Ω, a system of homogeneous equations is obtained in the powers of the constituent lens elements of the doublet, where the elements of the coefficient matrix η are the dispersion coefficients of the optical glasses comprising the doublet. Thus,

    ηΦ=0,                                              (36)

which has a nontrivial solution if and only if the determinant of the η-dispersion coefficient matrix vanishes, i.e., if and only if the determinant ##EQU15## This condition must be satisfied by the pair of glasses comprising the doublet, if axial color correction at three wavelengths is to be achieved by the thin lens doublet design. Thus, a particular pair of optical glasses can be used for designating a two-glass three-color lens doublet, if and only if the ratios of the primary dispersion coefficient to the secondary dispersion coefficient for both glasses are substantially equal, i.e., if and only if

    η.sub.11 /η.sub.21 =η.sub.12 /η.sub.22.    (38)

Since η_(ij) =ν_(ij) /[N(λ)-1], equation (37) implies that the determinant of the matrix of the corresponding ν-dispersion coefficients also vanishes, i.e., that ##EQU16## Thus, using the ν-dispersion coefficients, the condition that must be satisfied by a pair of optical glasses in order for those glasses to be compatible with each other in a two-glass three-color lens doublet can be expressed in terms of the ratios of the primary to secondary ν-dispersion coefficients as, viz.,

    ν.sub.11 /ν.sub.21 =ν.sub.12 /ν.sub.22.        (40)

Equation (36) indicates that the dioptric powers of the individual lens elements comprising the two-glass three-color lens doublet are functions of the dispersion coefficients of the glasses from which the lens elements comprising the doublet are made. The solution of equation (36) can be expressed in terms of the elements of either the first-row vector or the second-row vector of the η matrix. In terms of the elements of the first-row vector,

    φ.sub.1 =+k.sub.1 η.sub.12                         ( 41)

and

    φ.sub.2 =-k.sub.1 η.sub.11,                        (42)

where k₁ is a proportionality factor. For a lens doublet having unit focal length at the base wavelength λ₀, which is the specified focal length of the two-glass three-color doublet for purposes of this analysis,

    k.sub.1 =1/(η.sub.12 -η.sub.11).                   (43)

Similarly, in terms of the elements of the second-row vector,

    φ.sub.1 =+k.sub.2 η.sub.22                         ( 44)

and

    φ.sub.2 =-k.sub.2 η.sub.21,                        (45)

where the proportionality factor k₂ is given by

    k.sub.2 =1/(η.sub.22 -η.sub.21)                    (46)

for the lens doublet having unit focal length at λ₀.

It is instructive to compare equations (41) and (42), which give the dioptric powers of the lens elements of a three-color lens doublet, with equations (28) and (29), which give the dioptric powers of the lens elements of a two-color lens doublet. In format, equations (41) and (42) are similar to equations (28) and (29). However, the values of the dispersion coefficients η_(ij) obtained from the quadratic dispersion model for the three-color doublet are in general different from the values of the dispersion coefficients η_(ij) obtained from the linear dispersion model for the two-color doublet.

Two-Glass n-Color Lens Doublet

The theoretical considerations presented above for selecting pairs of glasses that are compatible for designing two-color and three-color lens doublets can be generalized and extended mathematically to define the conditions that must be satisfied in order for a pair of glasses to be compatible for designing thin lens doublets color-corrected at n wavelengths, i.e., two-glass n-color lens doublets.

An n-color lens doublet has, by definition, a common paraxial focal point at wavelengths λ_(j) for j=1, 2, 3, . . . , n. For an n-color lens doublet of unit total dioptric power (i.e., Φ=1), at the base wavelength λ₀, the dioptric powers φ₁ and φ₂ of the individual lens elements comprising the doublet are related by the expressions

    φ.sub.1 +φ.sub.2 =1                                (47)

and

    DΦ=0,                                                  (48)

where ##EQU17## and 0 is a column null vector, where λ₁ <λ₂ <λ₃ < . . . <λ_(j-1) <λ_(j) < . . . <λ_(n). Equation (48), which represents a set of (n-1) homogeneous equations that are linear in φ₁ and φ₂, has a nontrivial solution if and only if the rank of the coefficient matrix D is less than 2. The rank of the coefficient matrix D can be determined by examining all possible determinants of order 2 that can be formed by fixing any two rows of the matrix D. If all possible determinants of order 2 vanish, the rank of D is less than 2 and a nontrivial solution of equation (48) exists.

Two-Glass Four-Color Lens Doublet

For a two-glass lens doublet that is color-corrected at four wavelengths (i.e., for n=4), submatrices A, B and C of order 2 obtained from the coefficient matrix D of equation (48) are given by ##EQU18## A necessary condition for the existence of a two-glass lens doublet color-corrected at four wavelengths is that the determinants of the submatrices A, B and C vanish identically, which implies that the relationships ##EQU19## must be satisfied by the pair of optical glasses comprising the doublet.

To relate equation (52) to the η-dispersion coefficients of the glasses, a cubic model of the dispersion polynomial is used. Thus, the submatrix A of equation (49) can be written in the form

    A=Ω.sub.A η                                      (53)

where ##EQU20## and ##EQU21## The elements of the first and second column vectors of the η matrix are the η-dispersion coefficients of the first and second optical glasses, respectively, comprising the doublet. The matrix elements η₁₁, η₂₁ and η₃₁ are called the primary, secondary and tertiary dispersion coefficients, respectively, of the first glass of the doublet; and the matrix elements η₁₂, η₂₂ and η₃₂ are called the primary, secondary and tertiary dispersion coefficients, respectively, of the second glass of the doublet.

In equation (53), the submatrix A may be considered as the product of the factors Ω_(A) and η. From linear algebra, the rank of the submatrix A, written R(A), is related to the ranks of its factors Ω_(A) and η by the expression

    R(A)≦Minimum [R(Ω.sub.A), R(η)].          (54)

Since the determinant of the submatrix A vanishes as a necessary condition for the existence of a lens doublet that is color-corrected at four wavelengths, the rank of A must be less than 2, i.e., R(A)<2. The color coordinates ω_(j) are distinct in value for each wavelength. Since the highest-order nonsingular submatrix obtainable from the elements of the matrix Ω_(A) is equal to 2, it follows that R(Ω_(A))=2. Since the ηn-dispersion coefficients are finite in value, R(η)≠0. Furthermore, since R(A)<2, it follows from equation (54) that R(η)<2. Since R(η) must have an integral value less than 2 and greater than zero, it therefore follows that R(η)=1.

The fact that the rank of η in equation (53) is equal to unity implies that ##EQU22## Thus,

    η.sub.11 /η.sub.21 =η.sub.12 /η.sub.22     ( 56)

    η.sub.11 /η.sub.31 =η.sub.12 /η.sub.32     ( 57)

    η.sub.21 /η.sub.31 =η.sub.22 /η.sub.32,    (58)

which is to say that a particular pair of optical glasses can be used in designing a two-glass lens doublet that is color-corrected at four wavelengths, if and only if the ratios of the primary, secondary and tertiary dispersion coefficients of the two glasses are related to each other in accordance with equations (56), (57) and (58).

Just as the submatrix A in equation (49) was related to the η-dispersion coefficients by equation (53), so also the submatrix B in equation (50) and the submatrix C in equation (51) can be related to the η-dispersion coefficients by corresponding equations having the form B=Ω_(B) η and C=Ω_(C) η. These equations for B and C differ from the corresponding equation for A only in the first factor. Thus, an analysis of the equations B=Ω_(B) η and C=Ω_(C) η in the same way as was done above for equation (53) leads to the same result as expressed in equations (56), (57) and (58).

To understand the conditions expressed by equations (56), (57) and (58) in geometrical terms, consider each type of optical glass available to the designer to be represented by a unique point in a three-dimensional space, called a glass space, which is defined by a coordinate system having the η-dispersion coefficients η₁, η₂ and η₃ as orthogonal axes. Two particular optical glasses G₁ and G₂ having coordinates G₁ (η₁₁, η₂₁, η₃₁) and G₂ (η₁₂, η₂₂, η₃₂), respectively, in the glass space satisfy the conditions of equations (56), (57) and (58), if and only if G₁ and G₂ are collinear with the origin of the coordinate system.

The various optical glasses available to the designer can be represented as position vectors in η-space. The conditions of equations (56), (57) and (58), which must be satisfied in designing a two-glass lens doublet that is color-corrected at four wavelengths, require that the two glasses G₁ and G₂ selected for making the lens elements of the doublet must satisfy the vector equation

    G.sub.1 =αG.sub.2,                                   (59)

where G₁ and G₂ are the position vectors for the respective glasses, and α is a real scalar. Since the vectors G₁ and G₂ are collinear, the projections of G₁ and G₂ on the three orthogonal planes intersecting at the origin of the coordinate system must also be collinear, which is the geometric interpretation of the ratios given in equations (56), (57) and (58).

Two-Glass n-Color Lens Multiplet

In a two-glass lens multiplet, there may be any specified number of lens elements, but at least one of the lens elements is made of one type of glass and at least one other of the lens elements is made of another type of glass. A two-glass lens doublet is a special case of a two-glass lens multiplet in which there are only two lens elements.

In designing a two-glass n-color lens doublet, an (n-1)th degree glass dispersion model is used, which is to say that the power series in Buchdahl's dispersion equation (5) is approximated by an (n-1)th degree polynomial when axial color correction at n wavelengths is desired. In vector representation, each type of optical material available to the designer is represented as a unique glass point in an (n-1)-dimensional vector space having the dispersion coefficients η₁, η₂, η₃, . . . , η_(n-1) as orthogonal axes. Any two optical materials whose glass points are collinear with the origin of the (n-1)-dimensional vector space could in principle be used for designing a two-glass lens doublet that is color-corrected at n wavelengths.

Since axial color correction of a lens multiplet is dependent upon the dispersion properties of the glasses from which the lens elements comprising the multiplet are made, and does not depend upon the design form (i.e., the number, configuration and arrangement) of the individual lens elements, the technique described above for designing a two-glass n-color lens doublet can be extended for designing a two-glass n-color lens multiplet comprising more than two lens elements. For purposes of axial color correction, a two-glass lens multiplet can be treated mathematically as a quasi-doublet, where all the individual lens elements made of one type of glass are considered collectively as the first lens element of the quasi-doublet, and where all the individual lens elements made of the other type of glass are considered collectively as the second lens element of the quasi-doublet. Each of the two lens elements of the quasi-doublet is treated as having a dioptric power that is the sum of the dioptric powers of all the individual lens elements made of the same type of glass. Thus, in the mathematical analysis of a two-glass lens triplet, quadruplet, or higher-number multiplet treated as a quasi-doublet, at least one of the lens elements of the quasi-doublet actually comprises two or more individual lens elements made of the same type of glass. The total dioptric power of the quasi-doublet, of course, is the same as the total dioptric power obtained by summing the dioptric powers of the individual lens elements comprising the multiplet.

Since correction for axial chromatic aberration does not depend upon the design form of an optical system, the degree of axial color correction that can be obtained for an optical system using a particular pair of optical glasses cannot be varied by changing the design form of the system. Thus, the same degree of axial color correction can be obtained for any lens multiplet using two given optical glasses, regardless of the number of lens elements comprising the multiplet, and regardless of the configurations and dispositions of the lens elements. However, the remaining optical aberrations, including lateral color, the monochromatic aberrations, and the chromatic variations of the monochromatic aberrations, are sensitive to the design form of a lens system.

The selection of compatible optical materials is a necessary condition for designing an optical system that is color-corrected at a desired number of wavelengths. However, it is not sufficient merely to select compatible optical materials. The designer, working with compatible optical materials, must further apply his craft with skill in order to determine a design form that will enable the optical system to meet its specified requirements.

A designer can select a compatible pair of optical materials from among all the various types of optical materials available to him for designing a two-glass n-color lens multiplet by:

(a) Determining the dispersion coefficients for each type of optical material, the dispersion coefficients being coordinates defining a unique "glass point" for each type of optical material in an (n-1)-dimensional "glass space" coordinate system, where n is the number of wavelengths for which color correction is desired;

(b) Calculating the slope of the hyperline connecting the origin of the "glass space" coordinate system with the "glass point" for each type of optical material;

(c) Sorting the various types of optical materials in ascending order of the slopes of the hyperlines connecting the "glass points" with the origin of the coordinate system; and

(d) Selecting a pair of optical materials for which the hyperline slopes are substantially equal.

After identifying the various pairs of optical materials that are suitable in principle for designing the two-glass n-color lens multiplet, the designer then selects a preferred pair (or pairs) of optical materials to use in implementing the design. Ordinarily, it would be desirable for the dioptric powers of the individual lens elements comprising the lens multiplet to be minimized, which occurs when the absolute value of the difference between corresponding primary dispersion coefficients of the two glasses from which the lens elements are made is a maximum. A further step in the glass selection process, therefore, would ordinarily be to calculate the value of η₁₂ -η₁₁ for each pair of glasses that could in principle be used for designing the lens multiplet. The preferred pair of glasses is a pair for which the difference |η₁₂ -η₁₁ | is a maximum.

A particular pair of optical glasses selected according to the above-described procedure for designing a two-glass three-color lens doublet might also be found to provide color correction at more than three wavelengths. This result can occur, because the conditions for three-color correction comprise a subset of the conditions for color correction at more than three wavelengths. In general, if two glass vectors G₁ and G₂ are collinear, i.e., if the glass points G₁ and G₂ lie on the same hyperline in (n-1)-dimensional η-space, the projections of the vectors G₁ and G₂ on the hyperplane of the next lower degeneracy, i.e., the projections of G₁ and G₂ on the hyperplane in (n-2)-dimensional η-space, must also be collinear with respect to the origin.

Introduction of Lens Thickness

The foregoing mathematical analysis was developed using the thin lens equations (13) to (17), which represent an idealization that ignores the effect of lens thickness. In the real world, after a compatible pair of optical materials has been selected for designing a two-glass color-corrected lens multiplet, the individual lens elements to be made from those optical materials must be given suitable thicknesses in order to be constructible.

In accordance with the present invention, an algorithm has been developed for introducing lens thickness into the design of a two-glass lens multiplet whose "thin lens" elements are to be made from optical glasses selected according to the above-described procedure. The algorithm for lens thickening requires that the powers of the "thin lens" elements be unchanged during the introduction of lens thickness. Also, in order for the "thickened" multiplet to have a total dioptric power (or focal length) that is identically equal to the total dioptric power (or focal length) of the idealized thin lens multiplet that it replaces, the second principal plane of any given lens element of the multiplet must coincide with the first principal plane of an adjacent lens element of the multiplet.

The algorithm for two-glass lens multiplet thickening is described in detail below for the special cases of a cemented lens doublet and an air-spaced lens doublet.

Two-Glass Cemented Lens Doublet

The thick lens dioptric powers φ₁ and φ₂, respectively, of the lens elements comprising a thickened lens doublet are given by the equations ##EQU23## where P_(j) (j=1, 2, 3, 4) are the refractive powers of the lens element surfaces in the doublet system, where t₁ and t₂ are the axial thicknesses of the respective lens elements of the system, and where N₁ and N₂ are the refractive indices of the respective glasses at the base wavelength λ₀.

The refractive power P_(j) of a surface of a lens element is a function of the surface curvatuve C_(j) (j=1, 2, 3, 4) and of the refractive index of the particular glass from which the lens element is made, according to the equations

    P.sub.1 =C.sub.1 (N.sub.1 -1)                              (62)

    P.sub.2 =C.sub.2 (1-N.sub.1)                               (63)

    P.sub.3 =C.sub.3 (N.sub.2 -1)                              (64)

    P.sub.4 =C.sub.3 (1-N.sub.2).                              (65)

The requirement that adjacent principal planes of the two lens elements of the doublet coincide implies that ##EQU24## which when solved for P₄ yields ##EQU25##

If the two lens elements are cemented together, C₂ =C₃. Thus, equations (63) and (64) can be combined and solved for P₃ to yield ##EQU26## Substituting the expressions for P₄ and P₃ from equations (67) and (68) into equation (61) yields

    φ.sub.2 =a.sub.1 P.sub.1 +a.sub.2 P.sub.2 +a.sub.3 P.sub.1 P.sub.2, (69)

where

    a.sub.1 =-(N.sub.2 t.sub.1 φ.sub.2)/N.sub.1 t.sub.2 φ.sub.1 ; a.sub.2 =(N.sub.2 -1)/(1-N.sub.1);

and

    a.sub.3 =(N.sub.2 -1)t.sub.1 φ.sub.2 /(1-N.sub.1)N.sub.1 φ.sub.1.

Equation (69) and the equations for a₁, a₂ and a₃ can be solved simultaneously to yield an equation

    AP.sub.2.sup.2 +BP.sub.2 +C=0,                             (70)

which is quadratic in P₂, where A=a₂ b-a₃ ; B=a₂ -a₁ +a₃ φ₁ -bφ₂ ; C=a₁ φ₁ -φ₂ ; and b=-t₁ /N₁. When the discriminant of the quadratic equation (70) is positive, there are two distinct values for P₂. Usually, only one of the two values for P₂ can result in a physically realizable lens system.

After the values of P₄, P₃ and P₂ have been determined, the values of P₁ can be found by simple algebraic substitution in equation (66). The values of the constructional parameters of the lens system are calculated from equations (62) to (65), after the values of the refractive powers P_(j) have been determined.

Two-Glass Air-Spaced Lens Doublet

The total dioptric power Φ of an air-spaced lens doublet is

    Φ=φ.sub.1 +(1-k)φ.sub.2,                       (71)

where k=dφ₁, where d is the axial separation of the lens elements in air. The procedure used to calculate the dioptric powers φ₁ and φ₂ of the individual lens elements of the doublet is identical to the procedure used for the idealized case of two thin lenses in contact, except that total dioptric power Φ is given by equation (71) and the parameter k varies with wavelength according to the relationship

    k(λ)=k[1+D.sub.1 (λ)],                       (72)

where k is the value of k(λ) at the base wavelength λ₀.

Introduction of thickness into the design of an air-spaced lens doublet follows the general procedure described above for designing a cemented lens doublet. The individual lens elements are "thickened" by keeping the powers φ₁ and φ₂ the same, and by keeping the axial separation of the adjacent principal planes equal to d.

Two-Glass Mangin Mirror

The glass selection procedure described above is applicable to catadioptric systems (i.e., systems involving both reflection and refraction of light), as well as to dioptric systems involving only refraction of light. The simplest form of catadioptric system comprises a meniscus lens with spherical surfaces, one surface of which is provided with a reflective coating to form a mirror. Such a "lens mirror" is commonly called a Mangin mirror, after the French military officer who first proposed such imaging systems.

The classical Mangin mirror, which comprises only a negative singlet lens made of one optical material, is not corrected for primary chromatic aberration. In order for a Mangin mirror to be useful in imaging applications over a reasonable spectral range, it is necessary to provide color correction. The requirement for a color-corrected Mangin mirror led to the development of the so-called Mangin doublet mirror, which is a compound Mangin mirror in which the classical negative meniscus lens is replaced by a doublet lens system in which the two lens elements are made of different types of glasses.

Mangin doublet mirrors corrected for axial chromatic aberration at two wavelengths, like ordinary achromatic lens doublets (i.e., two-glass lens doublets corrected at two wavelengths), were known in the prior art. Designs for Mangin doublet mirrors color-corrected at two wavelengths were described by N. v. d. W. Lessing in Appl. Opt. 8, 979 (1969), and by S. Rosin and M. Amon in Appl. Opt. 6, 963 (1967) and in U.S. Pat. No. 3,494,688. A method for selecting pairs of optical glasses for designing cemented Mangin doublet mirrors that are color-corrected at two wavelengths was described by N. v. d. W. Lessing in Appl. Opt. 8, 1245 (1969). However, the prior art had no unified theory for selecting optical materials for designing compound Mangin mirrors color-corrected at more than two wavelengths.

Algorithms developed in the prior art for selecting optical materials to be used for designing two-glass Mangin mirrors failed to identify any combinations of glasses or other optical materials that could be used for designing any kind of catadioptric system capable of bringing more than two wavelengths to a common focus. The unified theory leading to the present invention, however, provides a method for selecting optical materials for designing Mangin mirrors that are color-corrected at more than two wavelengths.

Two-Glass Mangin Doublet Mirror

A thin cemented Mangin doublet mirror in air, having total catadioptric power Φ at base wavelength λ₀, can be considered as a system comprising two thin lenses and a mirror in contact. For a marginal paraxial ray trace through the system, the total catadioptric power Φ is therefore given by

    Φ=2φ.sub.1 +2φ.sub.2 +φ.sub.m,             (73)

where φ₁ and φ₂ are the respective dioptric powers of the thin lens elements and φ_(m) is the catoptric power of the mirror element comprising the system.

At a wavelength other than λ₀, the total catadioptric power Φ of a Mangin doublet mirror in terms of the dispersive powers D₁ (λ) and D₂ (λ) of the respective thin lens elements, as defined by equation (7), is given by

    Φ(λ)=2φ.sub.1 [1+D.sub.1 (λ)]+2φ.sub.2 [1+D.sub.2 (λ)]+φ.sub.m,                                  (74)

which after algebraic simplification becomes

    Φ(λ)=Φ+2φ.sub.1 D.sub.1 (λ)+2φ.sub.2 D.sub.2 (λ).                                               (75)

The difference in the total catadioptric power of a Mangin doublet mirror at the wavelengths λ_(i) and λ_(j), which is represented as ΔΦ(λ_(i),λ_(j))=Φ(π_(i))-Φ(.lambda._(j)), is given by

    ΔΦ(λ.sub.i,λ.sub.j)=2[φ.sub.1 D.sub.1 (λ.sub.i,λ.sub.j)+φ.sub.2 D.sub.2 (λ.sub.i,λ.sub.j)],                         (76)

where D₁ (λ_(i),λ_(j)) and D₂ (λ_(i),λ_(j)) represent the differences over the wavelength range from λ_(i) to λ_(j) between the dispersive powers of the respective glasses from which the lens elements comprising the Mangin doublet mirror are made.

Two-Glass Mangin Doublet Mirror Color-Corrected at Three Wavelengths

The conditions that must be satisfied for a two-glass Mangin doublet mirror to be color-corrected at three wavelengths λ₁, λ₂ and λ₃, where λ₁ <λ₂ <λ₃, imply that ΔΦ(λ₁,λ₂) and ΔΦ(λ₂,λ₃) must both be identically equal to zero. In matrix form, this can be written as

    DΦ=0,                                                  (77)

where ##EQU27## and ##EQU28## Equation (77) is identical to equation (30), which expresses the conditions that must be satisfied for a two-glass doublet lens system to be color-corrected at three wavelengths. Thus, as suggested above in the development of equations (31) to (38), in order for two particular optical materials to be suitable for designing a Mangin doublet mirror that is color-corrected at three wavelengths, it is necessary that the ratio of the primary dispersion coefficient to the secondary dispersion coefficient be substantially the same for both optical materials. Geometrically, the glass points for both optical materials in dispersion coefficient space must lie on a common line through the origin of the two-dimensional coordinate system defining the dispersion coefficient space.

The dioptric powers of the two thin lenses comprising the Mangin doublet mirror at the base wavelength λ₀ are given by

    φ.sub.1 =Kη.sub.12                                 ( 78)

and

    φ.sub.2 =-Kη.sub.11,                               (79)

where K is a proportionality factor. Equations (78) and (79) are correspondingly similar to equations (41) and (42), respectively; although the proportionality factor K of equations (78) and (79) is in general different from the proportionality factor k₁ of equations (41) and (42). For a Mangin doublet mirror having unit focal length (and therefore unit total dioptric power) at the base wavelength λ₀, equation (73) becomes

    2φ.sub.1 +2φ.sub.2 +φ.sub.m =1.                (80)

Substituting equations (78) and (79) into equation (80) yields ##EQU29## which enables equations (78) and (79) to be written in the form ##EQU30##

Equations (82) and (83) give the dioptric powers at λ₀ of the lens elements of the Mangin doublet mirror as a function of the η-dispersion coefficients of the lens materials and the catoptric power of the mirror element. The catoptric power φ_(m) of the mirror element is independent of wavelength variations, and can be set to a desired value by the optical designer. For any given value chosen for φ_(m), the powers φ₁ and φ₂ of the lens elements are inversely proportional to (η₁₂ -η₁₁). Thus, in order to minimize the values of φ₁ and φ₂, the glass pair used for designing the Mangin doublet mirror should be selected so that |η₁₂ -η₁₁ | is as large as possible.

In the special case where the mirror element of the Mangin doublet mirror is planar, i.e., where φ_(m) =0, equations (82) and (83) become ##EQU31## indicating that the dioptric power distribution of the lens elements of a Mangin doublet mirror that is color-corrected at three wavelengths has a value of one-half the dioptric power distribution of the same lens elements used for a two-glass lens doublet that is color-corrected at three wavelengths.

In the special case where the catoptric power of the mirror element of the Mangin doublet mirror is given by φ_(m) =-1, equations (82) and (83) reduce to ##EQU32## For this particular configuration of the mirror element, the dioptric power distribution of the lens elements of a Mangin doublet mirror that is color-corrected at three wavelengths is equal to the dioptric power distribution of the same lens elements used for a two-glass lens doublet that is color-corrected at three wavelengths.

Two-Glass Mangin Doublet Mirror Color-Corrected at n Wavelengths

The procedure for selecting two different optical materials for designing a Mangin doublet mirror that is color-corrected at n wavelengths is the same as the procedure described above for selecting two different optical materials for designing a lens doublet that is color-corrected at n wavelengths. Each optical material available to the designer is represented as a unique glass point G_(q) (q=1, 2) in an (n-1)-dimensional vector space, where the η-dispersion coefficients of each particular optical material are the coordinates of the glass point representing that particular optical material.

A pair of optical materials is suitable for designing a Mangin doublet mirror that is color-corrected at n wavelengths, if and only if the glass points G₁ and G₂ for the two materials obey the vector equation G₁ =αG₂. In comparing a Mangin double mirror color-corrected at n wavelengths with a lens doublet color-corrected at n wavelengths, where the same two types of glasses are used in making the Mangin doublet mirror and the lens doublet, the only difference appearing in the mathematical analysis of the system designs would be in the dioptric power distributions of the thin lens elements.

Two-Glass Mangin Multiplet Mirror Color-Corrected at n Wavelengths

The procedure described above for designing a two-glass Mangin doublet mirror color-corrected at n wavelengths can be generalized for designing a two-glass Mangin multiplet mirror color-corrected at n wavelengths. The two-glass Mangin multiplet mirror can be treated mathematically as a two-glass Mangin quasi doublet mirror, where all the individual lens elements made of one type of glass are considered collectively as the first lens element of the Mangin quasi-doublet mirror, and all the individual lens elements made of the other type of glass are considered collectively as the second lens element of the Mangin quasi-doublet mirror. The total dioptric power of the first lens element is the sum of the dioptric powers of all the individual lens elements made of one type of glass, and the total dioptric power of the other lens element is the sum of the dioptric powers of all the individual lens elements made of the other type of glass. The total catadioptric power Φ of the Mangin multiplet mirror is obtained by generalizing equation (73) into the form ##EQU33## where P_(i) (i=1, 2, . . . , r) represents the dioptric power of the ith lens element made of one type of glass, and Q_(j) (J=1, 2, . . . , s) represents the dioptric power of the jth lens element made of the other type of glass. The splitting of the powers of the lens elements of an optical system is a useful technique in optical design to reduce spherochromatism and the higher-order monochromatic aberrations.

Color-Corrected Anamorphic Optical Systems

Each of the optical systems described above is rotationally symmetrical about a principal axis, called the optical axis, of the system. However, the glass selection procedure of the present invention is generally applicable regardless of whether such symmetry exists, and hence is also applicable to anamorphic optical systems comprising, e.g., prisms.

For a thin prism of refractive index N₀ at the base wavelength λ₀ with apex angle α, the angle of deviation β₀ of a ray at wavelength λ₀ after passing through the prism is given by

    β.sub.0 =α[N.sub.0 -1].                         (89)

The angle of deviation β of a ray at wavelength λ after passing through the prism is given by

    β(λ)=α[N(λ)-1].                   (90)

From equation (6), N(λ)-N₀ =δN=(N₀ -1)D(λ), where D(λ) is the dispersive power of the glass from which the prism is made. Solving for [N(λ)-1] in terms of D(λ), and substituting α=β₀ /(N₀ -1) from equation (89), yields

    β(λ)=β.sub.0 [1+D(λ)].             (91)

In terms of the difference in the dispersive power D(λ_(i),λ_(j)) of the glass over the wavelength range from λ_(i) to λ_(j), as defined above in connection with equation (8), the chromatic dispersion of the prism from which the glass is made, i.e., Δβ(λ_(i),λ_(j))=β(λ_(i))-β(λ_(j)), is given by

    Δβ(λ.sub.i, λ.sub.j)=β.sub.0 D(λ.sub.i, λ.sub.j).                        (92)

Two-Glass Prism System Color-Corrected at Three Wavelengths

Consider a prism system comprising two thin prisms made of different optical materials having refractive indices N₁ and N₂, respectively, at the base wavelength λ₀, and having apex angles α₁ and α₂, respectively. The total angular deviation B of a ray passing through the prism system is the sum of the angular deviations produced by the individual prisms comprising the system. Thus, at the base wavelength λ₀, the total angular deviation is

    B=β.sub.1 +β.sub.2,                              (93)

where β₁ =α₁ (N₁ -1) and β₂ =α₂ (N₂ -1).

The conditions for the prism system to have zero chromatic dispersion at λ₁, λ₂ and λ₃, where λ₁ <λ₂ <λ₃, are that Δβ(λ₁,λ₂) and Δβ(λ₂,λ₃) both identically equal zero. From equation (92), these conditions can be written in matrix form as ##EQU34## Equation (94) is similar to equation (30) for a two-glass lens doublet color-corrected at three wavelengths, and to equation (77) for a two-glass Mangin doublet mirror color-corrected at three wavelengths. Using a quadratic dispersion model, the requirement for a particular pair of optical glasses to be suitable for designing a two-glass prism system, whose chromatic dispersion at wavelengths λ₁, λ₂ and λ₃ is zero, is that the determinant of the η-dispersion coefficients of the glasses vanish, i.e., ##EQU35## This implies that the individual thin prisms comprising the prism system must have angular deviations given by

    β.sub.1 =kη.sub.12                                ( 96)

and

    β.sub.2 =-kη.sub.11,                              (97)

where k is another proportionality constant. Substituting equations (96) and (97) into equation (93), yields ##EQU36## for the proportionality constant. Thus, equations (96) and (97) can be rewritten as ##EQU37## From equations (99) and (100), the apex angles α₁ =β₁ /(N₁ -1) and α₂ =β₂ /(N₁ -1) can be written as ##EQU38## Dividing equation (101) by equation (102) yields the ratio ##EQU39## Since η_(ij) =ν_(ij) /(N_(j) -1), equation (103) reduces to

    α.sub.1 /α.sub.2 =-ν.sub.12 /ν.sub.11,   (104)

which indicates that in order for a two-glass prism system to be color-corrected at three wavelengths, the ratio of the apex angles of the individual prisms comprising the system must be inversely proportional to the ratio of the ν-dispersion coefficients of the optical materials from which the prisms are made.

Equation (101) and (102) were derived using equations for thin prisms, and hence yield only approximate values. In designing a real prism system, the approximate values for α₁ and α₂ provided by thin lens theory can be adjusted by exact ray-tracing based on Snell's law.

Two-Glass Prism System Color-Corrected at More Than Three Wavelengths

Three-glass prism systems color-corrected at four wavelengths were described by R. E. Stephens in J. Opt. Soc. Am. 56, 213 (1966). However, the prior art did not have a unified theory for selecting optical materials from which such three-glass four-color prisms could be designed. No two-glass prism system color-corrected at more than three wavelengths has been reported in the prior art.

In accordance with the present invention, the procedure for selecting optical materials for designing a two-glass prism system that has zero chromatic dispersion at more than three wavelengths is the same as the procedure described above for selecting optical materials for designing a two-glass lens system or a two-glass catadioptric system that is color-corrected at more than three wavelengths.

The glass selection procedure described above is of general validity, and can be used for designing two-glass optical systems, e.g., lens systems, catadioptric systems and prism systems, in which color correction at three or more wavelengths is required. Any color-corrected optical system designed using the glass selection procedure described above would therefore be within the scope of the present invention.

Extention of Theory to Include Optical Systems Comprising More Than Two Glasses

Considering an optical multiplet comprising k thin lens elements in contact made from q different types of glasses, where k≧q, and where the multiplet is color-corrected at n wavelengths, the power series expansion in Buchdahl's dispersion equation (5) is approximated by an (n-1)th degree polynomial in the color coordinate ω. Thus, in equation (7), the terms higher than the (n-1)th degree term are neglected, so that the dispersive power D₁ (λ) for each glass q of the multiplet can be expressed as ##EQU40## where the dispersion coefficients η_(iq) in general vary from glass to glass. Using this (n-1)th degree polynomial dispersion model, each glass of the multiplet can be represented by the parameters N_(q) (λ₀), η_(1q), η_(2q), . . . , η.sub.(n-1)q, where N_(q) (λ₀) is the refractive index of the qth glass at the base wavelength λ₀.

Similarly, the difference in the dispersive power over a wavelength range λ₁ <λ<λ₂ for each glass q, represented by D_(q) (λ₁,λ₂)=D_(q) (λ₁)-D_(q) (λ₂), can be expressed in terms of the color coordinate ω by the power series ##EQU41##

It is noted that an [N_(q) (λ₀), ν_(1q), ν_(2q), . . . , ν.sub.(n-1)q ] glass parameter representation could be used instead of the [N_(q) (λ₀), η_(1q), η_(2q), . . . , η.sub.(n-1)q ] glass parameter representation. However, since the dioptric power of each lens element in the multiplet at λ₀ is directly proportional to [N(λ₀)-1], which is used in defining the η-dispersion coefficients, the η-dispersion coefficients generally lead to more compact and simpler algebraic formulations than the ν-dispersion coefficients, and are thus more desirable for rigorous mathematical analysis.

Algorithms for Determining Dispersion Coefficients

The dispersion coefficients η_(iq) for the various types of glasses listed in the catalogs of the major optical glass manufacturers can be calculated using either equation (105) or equation (106).

Since the dispersive power D₁ (λ) for a particular optical glass q at any wavelength λ can be calculated from ##EQU42## using data provided in the manufacturer's catalog for that particular glass, it is possible to form a system of (n-1) equations that are linear in the dispersion coefficients η_(iq) derived from equations (105) and (107), where the (n-1) equations correspond to (n-1) different wavelengths that are all different from the base wavelength λ₀. For each glass q, this system of nonhomogeneous linear equations can be written in matrix form as

    Ωη.sub.q =D.sub.q,                               (108)

where ##EQU43##

It is a property of equation (108) that the matrix Ω of the color coordinates ω can be expressed as the product of two square matrices of order (n-1), where one of the factors is a diagonal matrix and the other is a Vandermonde matrix. In factored, form, the matrix Ω can be written

    Ω=Diag(ω.sub.1, ω.sub.2, ω.sub.3, . . . , ω.sub.n-1)V(ω.sub.1, ω.sub.2, ω.sub.3, . . . , ω.sub.n-1),                                         (109)

where ##EQU44## The determinant of Ω is equal to the product of the determinants of its factors. Thus, ##EQU45## where II is the product symbol.

For a given selection of the base wavelength λ₀ and (n-1) wavelengths λ_(j), where λ_(j) ≠λ₀ and j=1, 2, . . . , (n-1), the corresponding color coordinates ω_(j), where j=1, 2, . . . , (n-1), are distinct and not equal to zero. This implies that the values of the determinants of the factors of Ω can never be zero. Thus, Ω is a nonsingular matrix with rank equal to its order. This further implies that equation (108) has a unique solution, and that the set of values of the dispersion coefficients η_(iq) for the glasses from which the lens elements comprising the multiplet are made is unique relative to the n wavelengths λ_(j).

According to the generalized algorithm, therefore, the (n-1)th degree polynomial as defined by equation (105) is equivalent to a polynomial fit of the dispersive power D_(q) (λ) of the optical glasses for (n-1) wavelengths λ_(j) relative to the base wavelength λ₀. The (n-1) wavelengths λ_(j) can be selected to be the particular wavelengths for which color correction is required, i.e., the wavelengths for which the multiplet is designed to have a common focus.

An alternative algorithm for obtaining a set of (n-1) dispersion coefficients n_(iq) characterizing the dispersion properties of an optical glass can be developed using equation (106) for the dispersive power differences relative to wavelength pairs. For each glass, n values of its refractive index at n different wavelengths λ_(j), where j=1, 2, . . . , n, are selected from the manufacturer's catalog. Assuming that λ_(j) ≠λ₀, and that λ₁ <λ₂ < . . . <λ_(n-1) <λ_(n), a system of (n-1) nonhomogeneous equations that are linear in the glass dispersion coefficients is obtained, from which the values of η_(iq) are calculated. This system of equations can be written in matrix form as

    Δωη.sub.q =ΔD.sub.q,                 (111)

where ##EQU46## This system of equations represented by the matrix equation (111) is different from the system of equations represented by the matrix equation (108) obtained using the first algorithm described above. In general, the dispersion coefficients obtained from equation (111) have values that are slightly different from the dispersion coefficients obtained from equation (108) for the same types of glasses. This second algorithm using equation (111) may be viewed as a polynomial fit of the dispersive power differences of the glasses in terms of (n-1) wavelength pairs, rather than as a polynomial fit of the dispersive powers in terms of (n-1) wavelengths as in the first algorithm using equation (108).

Dioptric Powers of Lens Elements

Considering an optical multiplet comprising k cemented thin lens elements in air, where the thin lens elements are made from q different types of glasses, where k≧q, the total dioptric power Φ of the multiplet at the base wavelength λ₀ is equal to the sum of the dioptric powers of the individual thin lens elements at the same base wavelength λ₀. Thus, ##EQU47## where φ_(i) represents the subtotal dioptric power of the group of lens elements made of the same ith type of glass. For a multiplet of unit total dioptric power at λ₀ and color-corrected at n wavelengths λ_(j) (j=1, 2, . . . , n), the subtotal dioptric powers φ_(i) at λ₀ are related by the expressions

    φ.sub.1 +φ.sub.2 + . . . +φ.sub.i + . . . +φ.sub.q =1 (113)

and

    ΔΦ(λ.sub.1,λ.sub.2)=ΔΦ(λ.sub.2,.lambda..sub.3)= . . . =ΔΦ(λ.sub.n-1,λ.sub.n)=0, (114)

where ##EQU48## In matrix form, equations (113) and (114) can be represented as

    GΦ=e.sub.1,                                            (116)

where ##EQU49##

A unique solution of equation (116) exists, if the determinant of the G matrix is not equal to zero. From equation (111), the G matrix can be written in terms of the dispersion coefficients η_(iq) and the color coordinates ω_(q) of the glasses. Expressing G as the product of two matrices, ##EQU50## where S=(1, 1, . . . , 1) is a sum vector (i.e., a vector having unity as a value for each component) of order q, and η is an "(n-1) by q" matrix in the dispersion coefficients given by and where ΔΩ is the matrix defined above by equation (111), and the 0 matrices are null matrices of order (n-1).

Two particular cases are of special interest in considering the dioptric power distribution of the lens elements, or subgroups of lens elements, of a color-corrected multiplet made of different types of glasses, viz., the case where n=q, and the case where n>q.

For the case where n=q, equation (116) represents a system of q equations that are linear in the unknown powers φ_(i) (i=1, 2, . . . , q). Since the number of equations is equal to the number of unknowns when n=q, a unique solution of equation (116) exists when the determinant of G is not equal to zero. The dioptric powers φ_(i) can then readily be calculated by using Cramer's Rule, or any of a number of other methods for solving systems of linear equations.

When G is written as the product of two matrices as in equation (117), the determinant of G is equal to the product of the determinants of its factors. Thus, ##EQU51## where ##EQU52##

By Cramer's Rule, the dioptric powers φ_(i) can be expressed as

    φ.sub.i =det M.sub.i /det G                            (120)

for i=1, 2, 3, . . . , n, where M_(i) is the matrix obtained from G by replacing the ith column of G by the unit column vector e₁.

In terms of the dispersion coefficients η_(iq) of the glasses, the matrix M_(i) can be written as the product of two matrices ##EQU53## where η_(i) is the matrix obtained from the dispersion coefficients matrix n when the ith column of η is replaced by a null column vector. Since ##EQU54## equation (120) becomes ##EQU55## where the matrix ##EQU56## is obtainable from ##EQU57## by replacing the ith column of ##EQU58## with the unit column vector e₁. Thus, ##EQU59##

The determinant of ##EQU60## when expanded in terms of the elements of the ith column and its cofactors, yields ##EQU61## where η_(i)(0) is a square matrix of order (n-1), which is obtained from the rectangular matrix η by deleting the ith column vector of η.

The determinant of ##EQU62## when expanded in terms of the elements of its first row and its cofactors, yields ##EQU63## Combining equations (125) and (123) yields ##EQU64## which expresses the dioptric powers φ_(i) of the lens elements, or groups of lens elements, in the multiplet as functions of the dispersion coefficients of the various types of glasses from which the lens elements are made.

Dioptric Powers of Two-Glass Lens Multiplet

In the simple case of a two-glass lens multiplet color-corrected at two wavelengths, i.e., where q=n=2, the dispersion model is linear and the η matrix in equation (117) reduces to η=[η_(ll) η₁₂ ]. Thus, η₁(0) =[η₁₂ ] and η₂(0) =[η₁₁ ], so that the dioptric powers φ_(i) (i=1, 2) for the two glasses, as given by equation (127), are ##EQU65## Equation (128) and (129) are thus seen to be identical to equations (28) and (29) derived above for the case of a two-glass doublet whose total power is equal to unity at the base wavelength.

Dioptric Powers of Three-Glass Lens Multiplet

It is convenient to treat a three-glass lens multiplet as a quasi-triplet (i.e., q=3) in which each effective lens element comprises all the individual lens elements of the multiplet made from one particular type of glass. A particular case of interest is a three-glass triplet color-corrected at three wavelengths (i.e., k=q=n=3), for which the dispersion model is quadratic, so that the η matrix in equation (117) assumes the form ##EQU66## From equation (130), the submatrices η₁(0), η₂(0) and η₃(0) reduce to ##EQU67## which when substituted into equation (127) yield the dioptric powers

    φ.sub.1 =(η.sub.12 η.sub.23 -η.sub.22 η.sub.13)/d (131)

    φ.sub.2 =-(η.sub.11 η.sub.23 -η.sub.21 η.sub.13)/d (132)

    φ.sub.3 =(η.sub.11 η.sub.22 -η.sub.21 η.sub.12)/d, (133)

where ##EQU68## i.e., d=η₁₁ (η₂₂ -η₃₃)+η₁₂ (η₂₃ -η₂₁)+η₁₃ (η₂₁ -η₂₂).

Four-Glass Lens Multiplet

Another particular case of interest is a four-glass lens quadruplet that is color-corrected at four wavelengths, i.e., k=q=n=4. Such a quadruplet was called a superachromatic quadruplet by N.v.d.W. Lessing in Appl. Opt. 9, 2390 (1970). Although various combinations of four different types of optical glasses suitable for designing such superachromatic quadruplets were reported in the prior art, a general technique for identifying all the optical materials that could be used for designing such superachromatic quadruplets was not developed until the present invention.

In accordance with the present invention, a cubic dispersion model is used to select glasses for designing a superachromatic quadruplet. Accordingly, the η matrix in equation (117) assumes the form ##EQU69## from which the submatrices η_(j)(0) are obtained, and the dioptric power distributions φ_(i) (=1, 2, 3, 4) are calculated from equation (127).

Although lens multiplets comprising q different types of optical glasses color-corrected at n=q wavelengths were known in the prior art for q=2, 3, 4, the method of the present invention whereby suitable optical materials for designing such multiplets can be identified by expressing the dispersion characteristics of the optical materials in terms of η-dispersion coefficients is completely novel. Furthermore, it is also a novel feature of the present invention that the practicality of using any particular combination of optical materials to design a lens multiplet can be determined a priori by expressing the dioptric powers of the individual lens elements of the multiplet as functions of the η-dispersion coefficients of the optical materials selected for making the lens elements.

Special Case for n>q

Equation (116), GΦ=e₁, represents a system of n equations that are linear in the unknown dioptric powers φ_(i) (i=1, 2, . . . , q). for the case where n>q, the number of equations would be greater than the number of unknowns. For k=q=2, i.e., for a two-glass doublet color-corrected at n wavelengths, equation (116) becomes ##EQU70## which is the matrix equation equivalent of equations (47) and (48) for a two-glass n-color lens doublet.

In general, for a lens multiplet made of q different types of glasses, the matrix D in equation (48) is an "(n-1)×q" rectangular matrix in D_(q) (λ_(j-1), λ_(j)). The conditions that must be satisfied by the individual glasses are obtained in the same way as for the doublet case, i.e., by examining the rank of the matrix D. Thus, a set of q glasses could be used for designing a lens multiplet that is color-corrected at n wavelengths, if and only if the rank of the matrix D for that set of glasses is less than q.

Special Case for n=q+1

An interesting special case, which is instructive in describing the invention, is a lens multiplet comprising q different types of optical glasses color-corrected at q+1 wavelengths. For this case, equation (116) becomes ##EQU71##

Equation (136) is equivalent to the sets of equations

    φ.sub.1 +φ.sub.2 + . . . +φ.sub.q =1           (137)

and

    DΦ=0,                                                  (138)

where ##EQU72## and 0 is a null column vector of order q.

In terms of the chromatic coordinates and the dispersion coefficients of the glasses, the homogeneous matrix equation (138) can be written as

    ΔΩηΦ=0,                                (139)

where ΔΩ is defined as in equation (111), and η is the matrix of the dispersion coefficients as defined in equation (117). Both ΔΩ and η are square matrices of order q. The matrix ΔΩ is nonsingular, so that equation (139) reduces to

    ηΦ=0.                                              (140)

In order for equation (140) to have a nontrivial solution, it is necessary that the determinant of η vanish, i.e., that ##EQU73## Thus, equation (141) represents the condition that must be satisfied by the η-dispersion coefficients of the q different types of glasses used in designing the lens multiplet in order for the multiplet to be color-corrected at q+1 wavelengths.

The general solution of the homogeneous equation (140) is

    φ.sub.j =αN.sub.ij                               ( 142)

for j=1, 2, . . . , q, where α is an arbitrary constant and the N_(ij) are the cofactors of the elements of the ith row of the η matrix. The constant α can be chosen to normalize the total dioptric power of the lens multiplet at the base wavelength λ₀. Substituting equation (142) into equation (137) yields ##EQU74##

The set of equations (142) and (143) enables the dioptric powers of the q different groups of lens elements corresponding to the q different types of glasses used in making the lens multiplet to be determined, where the lens multiplet is color-corrected at q+1 wavelengths. The above analysis for the case where n=q+1 holds true in the particular case where there is only one lens element in each of the groups of lens elements, i.e., where k=q.

When k=2, where k=q, there are only two lens elements, with each lens element being made of a different type of glass. Thus, equation (141) reduces to equation (37), and equation (142) reduces to equations (41) and (42), which were derived above for the case of a two-glass lens doublet color-corrected at three wavelengths.

When k=3, where k=q, the resulting lens multiplet is a three-glass triplet. In the case under consideration, the lens multiplet color-corrected at n=q+1 wavelengths becomes a three-glass lens triplet color-corrected at four wavelengths, for which the designation "superachromat" was used by Herzberger in U.S. Pat. No. 3,395,962. The present invention provides a technique for identifying all possible glass combinations that could be used in designing such superachromatic lens triplets, and for selecting practical three-glass combinations in terms of dioptric power distributions.

For k=q=3, equation (141) reduces to ##EQU75## and the dioptric powers of the individual lens elements, as given by equation (142), become

    φ.sub.1 =α.sub.1 (η.sub.12 η.sub.23 -η.sub.22 η.sub.13)                                             (145)

    φ.sub.2 =-α.sub.1 (η.sub.11 η.sub.23 -η.sub.21 η.sub.13)                                             (146)

    φ.sub.3 =α.sub.1 (η.sub.11 η.sub.22 -η.sub.21 η.sub.12),                                            (147)

where

    α.sub.1 =1/[η.sub.11 (η.sub.22 -η.sub.23)+η.sub.12 (η.sub.23 -η.sub.21)+η.sub.13 (η.sub.21 -η.sub.22)].

The expressions for the dioptric powers φ₁, φ₂ and φ₃ given in equations (145), (146) and (147) are identical to those in equations (131), (132) and (133) for the case of a three-glass lens triplet color-corrected at three wavelengths. However, a three-glass lens triplet color-corrected at four wavelengths must satisfy the additional conditions implied in equation (144). The conditions of equation (144) can be interpreted geometrically by considering each glass available to the designer as a unique point in a three-dimensional glass space defined by the axes η₁, η₂ and η₃. Three glasses, represented by the points G₁, G₂ and G₃, satisfy the conditions of equation (144), if and only if G₁, G₂ and G₃ lie in a common plane that passes through the origin of the coordinate system.

Applying a series of elementary row and column operations to the matrix of equation (144) results in ##EQU76## Defining μ₁ =η₁ /η₃ and μ₂ =η₂ /η₃, so that μ_(1k) =η_(1k) /η_(3k) and μ_(2k) =η_(2k) /η_(3k) for k=1, 2, 3, the parameters μ₁ and μ₂ can be considered as axes of a two-dimensional rectangular coordinate system into which the three-dimensional coordinate system defined by the axes η₁, η₂ and η₃ is transformed. Equation (148) implies that when the various optical materials available to the designer are plotted as points in the two-dimensional coordinate system with axes μ₁ and μ₂, the glass points G₁ (μ₁₁, μ₂₁), G₂ (μ₁₂, μ₂₂) and G₃ (μ₁₃, μ₂₃) are collinear. This two-dimensional geometrical representation of the glasses in (μ₁,μ₂) space is more convenient than the three-dimensional representation in (η₁, η₂, η₃) space. With the aid of a digital computer, it is relatively easy to obtain the coordinates for the glasses in (μ₁, μ₂) space from the known values of the n-dispersion coefficients of the optical materials in (η₁, η₂, η₃) space.

When k=4, where k=q, the resulting lens multiplet is a four-glass quadruplet. The case under consideration of a lens multiplet color-corrected at n=q+1 wavelengths becomes a four-glass lens quadruplet color-corrected at five wavelengths. The dispersion model is quartic, and equation (141) becomes ##EQU77## Equation (149) expresses the conditions that must be satisfied by a set of four optical materials in order for those four optical materials to be compatible with each other for use in designing a five-color lens quadruplet. The dioptric powers of the optical materials selected for designing the five-color lens quadruplet can be calculated from equation (142).

Geometrically, equation (149) implies that if each optical material available to the designer is represented by a unique point in a four-dimensional n-dispersion coefficient space with rectangular coordinate axes η₁, η₂, η₃ and η₄, the glass points G₁, G₂, G₃ and G₄ must lie in a common hyperplane that passes through the origin of the coordinate system.

A more easily visualizable geometrical interpretation of equation (149) can be obtained by applying a series of elementary row and column operations to the matrix of equation (149), which results in the transformed equation ##EQU78## where μ_(1k) =η_(1k) /η_(4k), μ_(2k) =η_(2k) /η_(4k) and μ_(3k) =η_(3k) /η_(4k) for k=1, 2, 3, 4.

It is noted that the definition of the parameter μ_(ik) (i=1, 2, 3) for the case of a four-glass five-color lens quadruplet is different from the definition of the parameter μ_(i) (i=1, 2) introduced above for the case of a three-glass four-color lens triplet.

Equation (150) implies that if each optical material available to the designer is represented by a unique point in a three-dimensional rectangular coordinate system with axes μ₁, μ₂ and μ₃, the four glasses with coordinates G₁ (μ₁₁, μ₂₁, μ₃₁), G₂ (μ₁₂, μ₂₂, μ₃₂), G₃ (μ₁₃, μ₂₃, μ₃₃) and G₄ (μ₁₄, μ₂₄, μ₃₄) must be collinear in the three-dimensional (μ₁, μ₂, μ₃) space in order to be suitable for designing a four-glass five-color lens quadruplet. With the aid of a digital computer, it is relatively easy to obtain the coordinates of the optical materials in the three dimensional (μ₁, μ₂, μ₃) space from the known values of the η-dispersion coefficients of those materials in the four dimensional (η₁, η₂, η₃, η ₄) space.

Method for Selecting Three Compatible Glasses

In selecting a combination of three different optical materials for designing a "three-glass" multiplet (i.e., an optical system made of three different optical materials, whether of glass or otherwise) that is color-corrected at four wavelengths, it is more convenient to use a two-dimensional glass point representation (μ₁, μ₂) space rather than a three-dimensional glass point representation in (η₁, η₂, η₃) space for each of the available optical materials.

The selection process starts with the choice of a particular optical material represented by glass point G₁ (μ₁₁, μ₂₁). The origin of the coordinate system defined by the axes μ₁ and μ₂ is then translated to G₁, so that the "new" axes μ₁ ' and μ₂ ' (i.e., the axes after translation) are parallel to the original axes μ₁ and μ₂, respectively. The coordinates of each optical material are then expressed in terms of the translated axes μ₁ ' and μ₂ ', so that G_(k) (μ_(1k), μ_(2k)) is expressed as G_(k) '(μ_(1k) ', μ_(2k) '), where ##EQU79## which are the equations for transformation of coordinates.

The equation of a straight line through the origin G₁ '(0,0) of the transformed coordinate system defined by the translated axes μ₁ ' and μ₂ ' is given by μ₂ '=m'μ₁ ', where m' is the slope of the line referred to the translated axes. Thus, a straight line passing through G₁ '(0,0) and a glass point G_(k) '(μ_(1k) ', μ_(2k) ') has a slope

    m'=μ.sub.2k '/μ.sub.1k ',                            (152)

which is the equation used to determine the slope m' for each available optical material. The arc tangent of m' is the slope angle of the glass point G_(k) ' relative to the μ₁ ' axis.

Using equation (152), a table is then made in which all available optical materials are sorted in ascending or descending order of the values of m'. Any pair of optical materials having substantially the same value of m' (or, equivalently, having substantially the same value of the arc tangent of m') as the originally chosen optical material, whose glass point G₁ is the origin of the translated axes, can be used in combination with that originally chosen optical material for designing a "three-glass" optical system that is color-corrected at four wavelengths.

Method for Selecting Four Compatible Glasses

In selecting a combination of four different optical materials for designing a "four-glass" multiplet that is color-corrected at five wavelengths, it is more convenient to use a three-dimensional glass point representation in (μ₁, μ₂, μ₃) space rather than a four-dimensional glass point representation in (η₁, η₂, η₃, η₄) space for each of the available optical materials.

The selection process starts by choosing two optical materials represented by glass points G₁ (μ₁₁, μ₂₁, μ₃₁) and G₂ (μ₁₂, μ₂₂, μ₃₂), respectively. The origin of the coordinate system defined by the orthogonal axes μ₁, μ₂ and μ₃ is then translated to G₁, so that the translated axes μ₁ ', μ₂ ' and μ₃ ' are parallel to the original axes μ₁, μ₂ and μ₃, respectively. The coordinates of each optical material are then expressed in terms of the translated axes, so that G_(k) (μ_(1k), μ_(2k), μ_(3k)) is expressed as G_(k) '(μ_(1k) ', μ_(2k) ', μ_(3k) '), where ##EQU80##

The equation of a straight line through the origin G₁ '(O, O, O) of the transformed coordinate system defined by the translated axes μ₁ ', μ₂ ' and μ₃ ' is given by μ₁ '/μ₁₂ '=μ₂ '/μ₂₂ '=μ₃ '/μ₃₂ ', which is equivalent to the system of equations ##EQU81## It is noted that any values for μ₁, μ₂ and μ₃ satisfying any two of the equations (154) also satisfy the third equation. Equations (154) also imply that ##EQU82## where k is a constant different from zero. Therefore, the glass points G₂ '(μ₁₂ ', μ₂₂ ', μ₃₂ '), G₃ '(μ₁₃ ', μ₂₃ ', μ₃₃ ') and G₄ '(μ₁₄ ', μ₂₄ ', μ₃₄ ') are collinear with the origin G₁ '(O, O, O), if and only if their corresponding coordinates are proportional to each other. Optical materials represented by glass points G₃ ' and G₄ ' can be selected from among those optical materials having coordinates that are substantially proportional to the corresponding coordinates of G₂ '.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a geometrical representation in dispersion coefficient space of an (n-1)th degree dispersion model for an optical system comprising two different optical materials.

FIG. 2 is a geometrical representation in dispersion coefficient space of a linear dispersion model for an optical system comprising two different optical materials.

FIG. 3 is a geometrical representation in dispersion coefficient space of a quadratic dispersion model for an optical system comprising two different optical materials.

FIG. 4 is a geometrical representation in dispersion coefficient space of a cubic dispersion model for an optical system comprising two different optical materials.

FIG. 5 is a geometrical representation in dispersion coefficient space of a cubic dispersion model for an optical system comprising three different optical materials.

FIG. 6A is a plot in the (η₁, η₂) plane of the η-dispersion coefficients for 813 different types of optical glasses listed in the catalogs of five major optical glass manufacturers.

FIG. 6B is an enlargement of a portion of the plot shown in FIg. 6A.

FIG. 7A is a plot in the (ν₁, ν₂) plane of the ν-dispersion coefficients for the same 813 different types of optical glasses whose η-dispersion coefficients are shown in FIG. 6A.

FIG. 7B is an enlargement of a portion of the plot shown in FIG. 7A.

FIG. 8 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a two-color cemented lens doublet having unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements comprising the doublet are made of Schott BK7 and Schott F2 glasses.

FIG. 9 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a three-color cemented lens doublet of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements comprising the doublet are made of Schott FK51 and Schott K51 glasses.

FIG. 10 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color cemented lens doublet of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements comprising the doublet are made of Corning-France 486817 and Ohara LAK14 glasses.

FIG. 11 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color air-spaced lens doublet of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements comprising the doublet are made of the same types of glasses as for the lens doublet of FIG. 10.

FIG. 12 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a five-color air-spaced lens doublet of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements comprising the doublet are made of Schott PK51 and Hoya LAC7 glasses.

FIG. 13 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color cemented two-glass lens triplet having a focal length that is 40 times the focal length of the doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10, where the first and third lens elements are made of Corning-France 486817 glass and the second lens element is made of Corning-France 589612 glass.

FIG. 14 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color two-glass lens triplet having a focal length that is 40 times the focal length of the doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements are made of the same types of glasses as for the lens triplet shown in FIG. 13, the first and second lens elements are cemented together, and an air gap is provided between the second and third lens elements.

FIG. 15 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color cemented two-glass lens quadruplet having a focal length that is 40 times the focal length of the doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10, where the first and third lens elements are made of Corning-France 486817 glass, and the second and fourth lens elements are made of Corning-France 589612 glass.

FIG. 16 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color two-glass lens quadruplet having a focal length that is 40 times the focal length of the doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements are made of the same types of glasses as for the quadruplet shown in FIG. 15, except that the first and fourth lens elements are made of Corning-France 486817 glass, and the second and third lens elements are made of Corning-France 589612 glass, where the first and second lens elements are cemented together, the third and fourth lens elements are cemented together, and an air gap is provided between the second and third lens elements.

FIG. 17 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color two-glass lens quadruplet having a focal length that is 40 times the focal length of the lens doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10, where the lens elements are made of the same types of glasses as for the quadruplet show in FIG. 15, where the first and third lens elements are made of Corning-France 486817 glass, and the second and fourth lens elements are made of Corning-France 589612 glass, where the first and second lens elements are cemented together, the third and fourth lens elements are cemented together, and an air gap is provided between the second and third lens elements.

FIG. 18 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color collimator comprising a two-glass lens quadruplet, where the two glasses from which the lens elements of the quadruplet are made of Hoya FCD10 glass and Ohara LAK14 glass.

FIG. 19 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a five-color collimator comprising a two-glass lens quadruplet, where the two glasses from which the lens elements of the quadruplet are made are Schott PK51 glass and Hoya LAC7 glass.

FIG. 20 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a five-color collimator comprising a lens quadruplet made of one type of optical glass and one type of crystal, where the glass is Hoya LAC7 glass and the crystal is calcium fluoride.

FIG. 21 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a five-color two-glass cemented Mangin doublet mirror system of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, where the system elements comprise a first lens element made of Hoya FDC10 glass and a second lens element made of Hoya FC3 glass the outward facing surface of the second lens element having a reflective coating.

FIG. 22 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color two-glass cemented Mangin doublet mirror system of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, where the system elements comprise a first lens element made of Hoya BAF11 glass and a second lens element made of Hoya FL6 glass, the outward facing surface of the second lens element having a reflective coating.

FIG. 23 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color two-glass cemented Mangin doublet mirror system of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, where the system elements comprise a first lens element made of Corning-France 486817 glass and a second lens element made of Corning-France 697554 glass, the outward facing surface of the second lens element having a reflective coating.

FIG. 24 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a five-color two-glass air-spaced Mangin doublet mirror system of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, where the system elements comprise a first lens element made of Schott PK51 glass and a second lens element made of Hoya LAC7 glass, the outward facing surface of the second lens element having a reflective coating.

FIG. 25 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a five-color two-glass air spaced Mangin doublet mirror system of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, where the system elements comprise a first lens element made of Hoya FCD10 glass and a second lens element made of Ohara PSK1 glass, the outward facing surface of the second lens element having a reflective coating.

FIG. 26 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-color two-glass air spaced Mangin doublet mirror system of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, where the system elements comprise a first lens element made of Hoya FCD10 glass and a second lens element made of Hoya LAC14 glass, the outward facing surface of the second lens element having a reflective coating.

FIG. 27 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a two-glass five-color Petzval lens system having a focal length of 600 mm at λ₀ =0.54607 μm for a relative aperture of f/3.5 and an angular field of ±6°.

FIG. 28 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a two-glass four-color Petzval lens system having a focal length of 360 mm at λ₀ =0.54607 μm for a relative aperture of f/3.5 and an angular field of ±6°.

FIG. 29 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a two-glass three-color Petzval lens system having a focal length of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/2 and an angular field of ±2.1°.

FIG. 30 provides a profile drawing and a plot of change in total deviation angle versus wavelength for a two-glass five-color doublet prism system having a total deviation angle of 6°.

FIG. 31 provides a profile drawing and a plot of change in total deviation angle versus wavelength for a two-glass five-color quintuplet prism system having a total deviation angle of 30°, where the apex angles of the first, second, third and fourth prism elements are equal to each other in absolute value, and where the apexes of the second and fourth prism elements have an orientation that is opposite to the orientation of the apexes of the first and third prism elements.

FIG. 32 provides a profile drawing and a plot of change in total deviation angle versus wavelength for a two-glass five-color quintuplet prism system having a total deviation angle of 30°, where the prism elements are made of the same types of glasses as for the corresponding prism elements of the prism system shown in FIG. 31, but where the apex angles of the five prism elements are all different in value from each other.

FIG. 33 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a three-glass lens quintuplet that is color-corrected at five wavelengths, the quintuplet having a focal length of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/10, and comprising a first lens element made of Ohara FKO1 glass, a second lens element made of Ohara KZFS8 glass and air-spaced from the first lens element, a third lens element made of Ohara FKO1 glass and cemented to the second lens element, a fourth lens element made of Ohara SF15 glass and air-spaced from the third lens element, and a fifth lens element made of Ohara KZFS8 glass and cemented to the fourth lens element.

FIG. 34 provides a profile drawing and a plot of paraxial marginal ray height versus wavelength for a four-glass lens quintuplet that is color-corrected at five wavelengths, the quintuplet having a focal length of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/10, and comprising a first lens element made of Schott PK51 glass, a second lens element made of Schott LAK20 glass and air-spaced from the first lens element, a third lens element made of Schott PK51 glass and cemented to the second lens element, a fourth lens element made of Schott KZFSN4 glass and air-spaced from the third lens element, and a fifth lens element made of Schott KSFS6 glass and cemented to the fourth lens element.

BEST MODE OF CARRYING OUT THE INVENTION

It is a feature of the present invention that combinations of optical glasses, which are compatible with each other for designing optical systems that are color-corrected at three or more wavelengths, can be readily selected from among the various optical glasses available to the designer.

Commercially available types of optical glasses are listed in catalogs published by various manufacturers. The latest edition (1980 Edition) of the Schott catalog lists 238 different types of optical glasses. The latest Ohara catalog (distributed in 1981) lists 227 different types of optical glasses. The latest Hoya catalog (distributed in 1981) lists 206 different optical glass types. The latest Corning-France catalog (1980 Edition) lists 80 different optical glass types. The latest Chance Pilkington catalog (1980 Edition) lists 62 different optical glass types. These five manufacturers presently produce a total of 813 different types of optical glasses.

The initial problem in designing an optical system that is to be color-corrected at more than two wavelengths is to select a suitable combination of optical materials from among the various types of optical materials available to the designer. The present invention provides a unified procedure for selecting suitable combinations of optical materials for use in designing color-corrected optical systems. The procedure of the present invention can be conveniently illustrated in connection with the design of a two-glass lens doublet (i.e., a multiplet comprising two lens elements made from different types of glasses), which is color-corrected at n different wavelengths, where n>2. The principles discussed herein apply as well to the selection of glasses or other optical materials for designing color-corrected lens triplets and higher-number lens multiplets.

The number of possible pairs of glasses that can be formed from the 813 different types of optical glasses listed in the catalogs published by the leading optical glass manufacturers can be determined by calculating the combination of n things taken r at a time, i.e., C(n, r)=n!/r!(n-4)!, where n=813 and r=2. This number, which is independent of the order in which the glasses are selected, is 330,078. Thus, if a "cut-and-try" approach were used to identify pairs of optical glasses that are compatible with each other for designing a two-glass color-corrected lens doublet, more than 330,000 different combinations of glasses would have to be investigated.

Very few pairs of optical glasses suitable for designing a two-glass lens doublet that is color-corrected at three wavelengths have been reported in the literature; and those few suitable glass pairs that have been reported were discovered by cut-and-try techniques without the benefit of a unified theoretical basis for selecting the particular glasses. Until the present invention, there has been no systematic way of knowing whether any particular glass could be paired with any other particular glass to produce an optical system that is color-corrected at more than two wavelengths.

In FIG. 1, two optical materials G₁ and G₂ are represented by corresponding glass points G₁ [η₁₁, η₂₁, η₃₁, . . . , η.sub.(n-1)1] and G₂ [η₁₂, η₂₂, η₃₂, . . . , η.sub.(n-1)2 ] in an (n-1)-dimensional orthogonal coordinate system defining a so-called glass space. The coordinates of any particular optical material in the glass space are the η-dispersion coefficients of the optical material, as derived from a generalized (n-1)th degree dispersion model. The condition that must be satisfied in order for a two-glass optical system comprising these two particular optical materials G₁ and G₂ to be color-corrected at n wavelengths is that the vectors G₁ and G₂ in the glass space satisfy the vector equation G₁ =αG₂, where α is the real scalar quantity. In graphic terms, as illustrated in FIG. 1, the glass points G₁ and G₂ must be substantially collinear with the origin of the coordinate system.

For the special case of a two-glass optical system that is color-corrected at two wavelengths (i.e., where n=2), the dispersion model used to derive the η-dispersion coefficients is linear. As shown in FIG. 2, a two-glass two-color optical system can be designed using any pair of optical glasses, because the glass points for all optical glasses necessarily lie on the same straight line in the one-dimensional η-dispersion coefficient space of the linear dispersion model. Although, any two different optical glasses could in principle be used in combination with each other for designing an optical system that is color-corrected at two wavelengths, it would normally be desirable to use only those pairs of glasses for which the absolute value of the difference between their corresponding primary dispersion coefficients is as large as possible.

For the special case of a two-glass optical system that is color-corrected at three wavelengths (i.e., where n=3), the dispersion model used to derive the η-dispersion coefficients is quadratic. As shown in FIG. 3, a two-glass three-color optical system can be designed using a pair of optical glasses whose glass points are collinear with the origin of the coordinate system defining the two-dimensional η-dispersion coefficient space of the quadratic dispersion model. Thus, if the glass points for the various optical glasses available to the designer are plotted on the (η₁, η₂) plane, suitable pairs of glasses that could be principle be used for designing an optical system that is color-corrected at three wavelengths can be ascertained by drawing straight lines from the origin of the coordinate system. Those glasses whose glass points are collinear with each other along any straight line through the origin of the coordinate system are in principle suitable candidates from which glass pairs can be selected for designing the two-glass three-color optical system.

Although any pair of glasses whose glass points are collinear with each other and with the origin of the coordinate system as shown in FIG. 3 could be used for designing a two-glass three-color optical system, it would ordinarily be desirable to select a pair of glasses from among those glasses satisfying the further condition that the absolute value of the difference between the corresponding primary dispersion coefficients is large. The particular design form of the optical system, including a determination of the appropriate number of lens elements (e.g., doublet, triplet, or higher-number multiplet), is determined in accordance with the principles of geometrical optics.

For the special case of a two-glass optical system that is color-corrected at four wavelengths (i.e., where n=4), the dispersion model used to derive the η-dispersion coefficients is cubic. As shown in FIG. 4, a two-glass four-color optical system can be designed using a pair of optical glasses whose glass points are collinear with the origin of the coordinate system defining the three-dimensional η-dispersion coefficient space of the cubic dispersion model. Thus, if the glass points for the various optical glasses available to the designer are plotted in the three-dimensional (η₁, η₂, η₃) space, suitable pairs of glasses that could in principle be used for designing an optical system that is color-corrected at four wavelengths can be ascertained by drawing straight lines from the origin of the coordinate system. Those glasses whose glass points are collinear with each other along any straight line through the origin of the coordinate system are in principle suitable candidates from which glass pairs can be selected for designing the two-glass four-color optical system. A preferred pair of glasses would ordinarily comprise glasses for which the absolute value of the difference between their corresponding primary dispersion coefficients is as large as possible.

The degree of the dispersion model used to derive the η-dispersion coefficients does not depend upon the number of glasses used in making the lens elements of the optical system. Therefore, the same three-dimensional plot of glass points shown in FIG. 4 in connection with the design of a four-color optical system comprising two different types of glasses could also be used for designing a four-color optical system comprising three different types of glasses. A three-glass lens system must necessarily comprise at least three lens elements. For a three-glass optical system that is color-corrected at four wavelengths (i.e., where n=4), the dispersion model used to derive the η-dispersion coefficients is cubic. As shown in FIG. 5, three glasses G₁, G₂ and G₃ are suitable for designing an optical system that is color-corrected at four or more wavelengths, if the glass points representing those three glasses in three-dimensional η-dispersion coefficient space are coplanar with each other and with the origin of the coordinate system.

In general, for an optical system that is color-corrected at n wavelengths, the dispersion model used to derive the η-dispersion coefficients is an (n-1)th degree polynomial, where n is the number of wavelengths for which color-correction is desired. Thus, a two-glass five-color optical system could be designed using any pair of glasses whose glass points lie on a common hyperline through the origin of an orthogonal coordinate system defining a four-dimensional hyperspace, where the axes of the coordinate system are the η-dispersion coefficients according to a fourth degree dispersion model. Likewise, a three-glass five-color optical system could be designed using any three glasses whose glass points line on a common hyperplane that passes through the origin of the coordinate system defining the same four-dimensional hyperspace.

In accordance with the present invention, an (n-1)th degree glass dispersion model is used to identify optical materials that are compatible with each other for designing an optical system color-corrected at n wavelengths. Each of the optical glasses available to the designer is represented by a unique point in an (n-1)-dimensional glass space, where the dispersion coefficients of the individual glasses are the coordinates of the glasses in the glass space. In practice, it is convenient to employ a digital computer to calculate and compare dispersion coefficient ratios for the various glasses available to the designer, where the dispersion coefficients are determined from a glass dispersion model in accordance with the number of wavelengths for which color correction is required for the particular application. For most color-correction requirements, it would not be necessary to use a glass dispersion model of order higher than cubic, i.e., n=4. For the analysis described herein, the η-dispersion coefficients are used, although the ν-dispersion coefficients could have been used.

In FIG. 6A, the glass points in η-dispersion coefficient space for 813 different optical glasses listed in the current catalogs of the five leading optical glass manufacturers are plotted in the two-dimensional η-dispersion coefficient space of the quadratic dispersion model. The values of η₁ and η_(`), which are called the primary and secondary η-dispersion coefficients, respectively, for the glasses according to a quadratic dispersion model, were calculated from equation (7) using a least-squares method for seven wavelengths in the visible region with a base wavelength of λ₀ =0.5740 μm. The seven wavelengths used were the solar emission lines 0.4047 μm, 0.4350 μm, 0.4861 μm, 0.5461 μm, 0.5876 μm, 0.6563 μm and 0.7065 μm.

The optical glasses whose glass point coordinates η₁ and η₂ are plotted graphically in FIG. 6A are also listed in TABLE I. The 813 glasses manufactured by the five leading optical glass manufacturers are identified in TABLE I by manufacturer's name, i.e., SCHOTT, OHARA, HOYA, Corning-France (abbreviated as CORN-F) and Chance Pilkington (abbreviated as CHANCE), and by catalog number. Also given for each glass listed in TABLE I are the base refractive index at λ₀ =0.574 μm, the primary dispersion coefficient η₁, the secondary dispersion coefficient η₂, the ratio (η₁ /η₂), and the arc tangent of the ratio (η₁ /η₂). The glasses are sorted in TABLE I in ascending order of the ratios (η₁ /η₂). The arc tangent of the ratio (η₁ /η₂) indicates the slope angle relative to the η₁ -axis, so that glasses having substantially the same arc tangent of the ratio (η₁ /η₂) are substantially collinear, and are therefore candidates for selection in forming pairs of glasses for designing optical systems that are color-corrected at three or more wavelengths. FIG. 6B provides an enlargement of a portion of the plot shown in FIG. 6A.

The glass selection procedure in accordance with the present invention can be illustrated graphically with reference to FIGS. 6A and 6B. In selecting a pair of glasses from among all the optical glasses whose glass points are plotted in FIG. 6A for designing a two-glass optical system that is color-corrected at three wavelengths, a designer first chooses one glass that is available to him and locates the glass point representing that one glass on the plot. The designer then draws a straight line extending through the glass point for that one glass and through the origin of the coordinate system. Other glasses that in principle are suitable for use in combination with that one glass for designing the desired two-glass three-color lens doublet lie substantially on the same straight line.

All glass points on the same straight line through the origin of the coordinate system have the same slope angle relative to the η₁ -axis. Therefore, a tabulation of the available optical glasses in terms of a convenient function of the slope angle, e.g., the arc tangent of the ratio (η₁ /η₂), as provided in Table I, can be used by an optical designer with greater convenience and accuracy than the graphical technique described above. With such a tabulation, all that the designer need do in selecting a pair of glasses for designing a two-glass three-color lens doublet is to choose a first glass, and then to identify other glasses whose tabulated values for the arc tangent of the ratio (η₁ /η₂) are substantially the same as for the chosen first glass. A suitable pair of glasses can be selected from among the glases having substantially the same value for the arc tangent of the ratio (η₁ /η₂).

Each of the glasses listed in TABLE I can be identified unambiguously by the manufacturer's catalog number, which might be an alphanumeric designation or a purely numeric designation depending upon the manufacturer. In a case where a new type of glass is manufactured by changing the constituents or percentage composition of an older type of glass, a new type of glass in invariably given a new catalog number by the manufacturer. Even where the new type of glass is made to meet the same optical specifications as an older type of glass, the new type of glass is given a new catalog number. The process of making any particular type of optical glass is ordinarily regarded by the manufacturer as proprietary information, and is ordinarily not known in detail by the optical designer who uses the glass. Accordingly, optical designers universally identify particular optical glasses by manufacturers' catalog numbers.

Considering the portion of the plot of FIG. 6A shown in enlarged view in FIG. 6B, it is instructive to identify glasses that could be paired with, for example, Hoya FCD10 glass for designing a two-glass optical system that is color-corrected at three wavelengths. Hoya FCD10 glass, with coordinates η₁ =-0.0610293 and η₂ =-0.00399528 in two-dimensional η-dispersion coordinate space (as ascertained by reference to page 10 of TABLE I), was selected for instructional purposes because the glass point for this particular optical glass is relatively isolated on the plot in FIG. 6B. A straight line, Line A, drawn through the origin of the coordinate system and the glass point for Hoya FCD10 glass indicates that e.g., Hoya FC3 glass (coordinates η₁ =-0.0842094 and η₂ =-0.00547060) can be combined with Hoya FCD10 glass for designing a two-glass optical system that is color-corrected at at least three wavelengths. An examination of the cubic and quartic dispersion models for these two glasses in the same spectral region as covered by TABLE I for the quadratic dispersion model shows that this particular pair of optical glasses can yield axial color correction at up to five wavelengths. This phenomenal compatibility of the Hoya FCD10 and Hoya FC3 glasses is discussed hereinafter in connection with a Mangin doublet mirror as illustrated in FIG. 21.

Line A in FIG. 6B also shows several other glasses that could be combined with Hoya FCD10 glass to form pairs of compatible glasses for designing optical systems that are color-corrected at at least three wavelengths. Thus, Schott BK8 glass (coordinates η₁ =-0.0870925 and η₂ =-0.00577220), Schott PSK50 glass (coordinates η₁ =-0.0823975 and η₂ =-0.00530444), Hoya PCD₂ glass (coordinates η₁ =-0.0878957 and η₂ =-0.00577861), Hoya ZNC7 glass (coordinates η₁ =-0.0905172 and η₂ =-0.00582287), and Chance Pilkington 641601 glass (coordinates η₁ =-0.0923210 and η₂ =-0.00588984) are also all substantially collinear with each other and with Hoya FCD10 glass and Hoya FC3 glass on Line A, and could in principle be considered along with Hoya FCD10 glass and Hoya FC3 glass as suitable candidates from which to select a pair of glasses for designing the desired two-glass three-color optical system. For optimum dioptric power distribution, however, it would usually be desirable to select a pair of glasses whose glass points are widely separated from each other along Line A, so that the difference between the η₁ primary dispersion coefficients for the two glasses is large.

As another example, it is instructive to consider the glasses that could be combined with Schott PK51 glass (coordinates η₁ =-0.0718368 and η₂ =-0.00220267) for designing a two-glass optical system that is color-corrected at at least three wavelengths. A line, Line B, drawn through the origin of the coordinate system and the glass point for Schott PK51 glass indicates that, e.g., Hoya LAC7 glass (coordinates η₁ =-0.0946047 and η₂ =-0.00290729 could be paired with Schott PK51 glass to form a pair of compatible glasses for designing an optical system that is color-corrected at at least three wavelengths. An examination of the cubic and quartic dispersion models for these two glasses in the same spectral region as covered by TABLE I for the quadratic dispersion model shows that this particular pair of glasses can be used for designing optical systems that are color-corrected at five wavelengths. This phenomenal compatibility of the Schott PK51 and Hoya LAC7 glasses is demonstrated hereinafter by the lens doublet shown in FIG. 12, the lens quadruplet shown in FIG. 19, the Mangin doublet mirror shown in FIG. 24, the Petzval lens system shown in FIG. 27, and the prism systems shown in FIGS. 30, 31 and 32.

A glance at Line B indicates that a number of other glasses, whose glass points lie substantially on Line B, are also suitable candidates for use in combination with Schott PK51 glass for designing an optical system that is color-corrected at three wavelengths, e.g., Schott K11 glass (coordinates η₁ =-0.0900976 and η₂ =-0.00275514), Schott LAK21 glass (coordinates η₁ =-0.0920718 and η₂ =-0.00285003), and Corning-France 511604 glass (coordinates η₁ =-0.0916603 and η₂ =-0.00283942).

It is important to remember that the values of η₁ and η₂ for the optical glasses listed in TABLE I were calculated using a least-squares method for seven selected wavelengths in the visible spectrum extending from a minimum wavelength of approximately 0.4 μm to a maximum wavelength of approximately 0.7 μm, with a base wavelength of 0.5740 μm. These wavelengths provide a statistically good representation of the entire visible spectrum. However, in a particular application, the designer might find it preferable to calculate values of η₁ and η₂ for the optical materials of interest by a least-squares method, or indeed by any other mathematical procedure, using different wavelengths. For example, if the designer were concerned with color-correction for infrared wavelengths, he could calculate a table of values for η₁ and η₂ for several wavelengths in the infrared wavelength band or bands of interest. If the designer were interested in obtaining color-correction at a number of different wavelength bands, as might be appropriate in designing, e.g., a color-corrected astronomical instrument having a common focal plane for radiation in the infrared and ultraviolet portions of the spectrum, he could calculate a table of values for η₁ and η₂ using several wavelengths in each of the wavelenth bands of interest.

The values calculated for η₁ and η₂, and therefore also the values calculated for the ratios (η₁ /η₂) and for the arc tangents of the ratios (η₁ /η₂), depend upon the wavelengths used in the quadratic dispersion model obtained from equation (7). In general, therefore, the values for η₁ and η₂ would differ from those shown in TABLE I, if different wavelengths were to be used in representing the spectral range (or the discontinuous spectral bands) of interest to the designer. The values listed in TABLE I, however, are useful in illustrating the invention, and furthermore are of practical utility in selecting pairs of compatible glasses for designing optical systems that are substantially color-corrected at at least three wavelengths in the visible spectrum.

In TABLE II, the same glasses manufactured by the same five optical glass manufacturers are again listed and unambiguously identified by manufacturer's name and catalog number, along with the base refractive index at λ₀ =0.589 μm for each glass. Also listed for each glass in TABLE II are the primary dispersion coefficient η₁, the secondary dispersion coefficient η₂, the teritary dispersion coefficient η₃, and the ratios (η₁ /η₂), (η₂ /η₃) and (η₁ /η₃), as calculated from equation (7) using a least-squares fitted cubic dispersion model. For each glass, the η-dispersion coefficients η₁, η₂ and η₃, as well as the base refractive index, were obtained for ten wavelengths in the extended spectral region from 0.3650 μm to 1.014 μm, with a base wavelength of λ₀ =0.589 μm. The glasses are sorted in TABLE II in ascending order of the ratios (η₁ /η₂). The ten wavelengths used were the solar emission lines 0.3650 μm, 0.4047 μm, 0.4350 μm, 0.4861 μm, 0.5461 μm, 5.5876 μm, 0.6563 μm, 0.7065 μm, 0.7682 μm and 1.014 μm.

Ordinarily, those optical glasses for which the corresponding ratios (η₁ /η₂), (η₂ /η₃) and (η₁ /η₃) are substantially equal to each other, and for which the glass points are substantially collinear with each other and with the origin of the coordinate system, are candidates for selection in forming pairs of compatible glasses to use for designing optical systems that are color-corrected at four wavelengths. It is sometimes more convenient to use the arc tangents of the ratios (η₁ /η₂), (η₂ /η₃) and (η₁ /η₃) rather than the ratios themselves in selecting compatible pairs of glasses for designing optical systems that are color-corrected at at least four wavelengths.

Since the η-dispersion coefficients and the ν-dispersion coefficients for any optical material are related by the equation η_(ij) =ν_(ij) /(N₀ -1), the selection of glass pairs for designing a two-glass three-color optical system could also be made from a plot of the glass points in ν-dispersion coefficient space, or from a tabulation of the values for the arc tangents of the ratios (ν₁ /ν₂), for the various optical glasses available to the designer. In FIG. 7A, in glass points are plotted in ν-dispersion coefficient space for the same 813 glasses whose glass points in η-dispersion coefficient space are plotted in FIG. 6A. An enlargement of a portion of the plot shown in FIG. 7A is shown in FIG. 7B.

To illustrate graphically the method of selecting a pair of compatible glasses for designing a two-glass three-color optical system using the ν-dispersion coefficients for the glasses available to the designer, consider Schott FK51 glass as the initial selection. Schott FK51 glass is represented in FIG. 7B by a glass point whose coordinates in ν-dispersion coefficient space are ν₁ =-0.031926 and ν₂ =-0.001819, as determined by the quadratic dispersion model used to plot the glass points shown in FIGS. 7A and 7B. A straight line, Line C, drawn through the origin of the coordinate system and the glass point for Schott FK51 glass indicates a number of other optical glasses whose glass points are substantially collinear with the origin and with the glass point for Schott FK51 glass. Examples of other suitable glasses lying on Line C include Schott K51 glass and Hoya ADF1 glass. All such glasses, whose glass points are substantially collinear with each other and with the origin, are suitable candidates from which to select a pair of glasses for designing an optical system having the required color correction at three wavelengths.

In FIG. 7B, straight lines could be drawn at various angles from the origin through the glass points representing the optical glasses available to the designer, thereby identifying suitable pairs of glasses for designing two-glass three-color optical systems. Thus, Line D in FIG. 7B indicates that, e.g., Corning-France 558525 glass and Schott BAF50 glass can be paired for designing a two-glass three-color optical system.

In FIG. 6A or FIG. 7A, any pair of optical glasses whose glass points lie substantially on the same straight line passing through the origin of the coordinate system can in principle be used for designing a two-glass three-color lens systems. Conversely, optical glasses whose glass points do not lie substantially on the same straight line cannot be used for designing a two-glass lens system that is color-corrected at three wavelengths. From equations (41 ) to (46), the dioptric powers φ₁ and φ₂ of the individual lens elements of a two-glass three-color lens doublet are inversely proportional to the absolute value of the difference between the primary dispersion coefficients η₁₁ and η₁₂ of the glasses used to make the lens elements of the doublet. It is therefore necessary that the absolute value |η₁₂ -η₁₁ | be as large as possible in order to minimize the dioptric powers of the individual lens elements. If the dioptric powers of the lens elements were too large, the configurations of the lens elements might be so rounded (resembling toy marbles in an extreme case) that it would be impractical to use them in most optical systems. The requirement that |η₁₂ -η₁₁ | be large would, therefore, ordinarily be treated as an additional condition that must be satisfied when selecting a pair of glasses for designing a color-corrected optical system.

In FIG. 6B or 7B, those optical glasses whose glass points are substantially collinear with each other and with the origin of the coordinate system are suitable candidates from which to select a pair of glasses for designing a two-glass optical system that is color-corrected at three or more wavelengths. Those glasses whose glass points are substantially collinear with the origin have ratios of primary to secondary dispersion coefficients that are substantially equal. Thus, in mathematical terms, when

    η.sub.11 /η.sub.21 =η.sub.12 /η.sub.22,

or, equivalently, when

    ν.sub.11 /ν.sub.21 =ν.sub.12 /ν.sub.22,

for two optical glasses, those two glasses in combination are suitable for designing a two-glass optical system that is color-corrected at three wavelengths. This property can readily be exploited with a digital computer by computing the slope of the line connecting each glass point with the origin, and then sorting all the glasses in ascending order of these "glass point" slopes.

It was known in the prior art that any two optical glasses can in principle be used in combination with each other for designing a lens doublet that is color-corrected at two wavelengths. This prior art knowledge is seen to be a special case of the unified theory upon which the present invention is based, where n=2. In FIG. 8, a cemented lens doublet of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10 is shown, where the lens elements comprising the doublet are made of Schott BK7 glass and Schott F2 glass. The two lens elements could be cemented together by any technique well-known in the art, as by using Canada balsam, optical cement or immersion oil. It can be seen by reference to page 9 of TABLE I that the ratio η₁ /η₂ =-13.6653 for Schott BK7 glass, and by reference to page 7 of TABLE I that the ratio η₁ /η₂ =-3.91263 for Schott F2 glass, are quite different in value. Thus, the unified theory upon which the present invention is based indicates that a lens doublet made of Schott BK7 and Schott F2 glasses cannot provide color correction at more than two wavelengths. Color correction is a function of the choice of glasses used for designing the doublet, and is independent of the particular design form of the doublet made from the chosen glasses. Thus, no amount of manipulation of the design form of a lens doublet made from these two glasses can result in color correction at more that two wavelengths.

It is a feature of the present invention that the suitability or non-suitability of any particular combination of optical materials for designing a color-corrected optical system can be readily determined by reference to tabulated data, such as by compairing tabulated values of the ratios η₁ /η₂ for a pair of optical materials being considered. The present invention eliminates the need for "cut-and-try" methods for determining whether any particular pair of optical materials can provide color correction at more than two wavelengths.

Although any two optical glasses are suitable in principle for designing a two-glass two-color lens doublet, there was no unified theory in the prior art for predicting whether any two particular optical glasses would have appropriate dioptric powers so that lens elements with practical configurations could be made from those particular glasses. It is a feature of the present invention, on the other hand, that the practicality or non-practicality of any two particular optical glasses for designing a lens doublet that is color-corrected at two or more wavelengths can be readily determined by reference to tabulated data.

The cemented lens doublet shown in FIG. 8, comprising lens elements made of Schott BK7 and Schott F2 glasses, is color-corrected at only two wavelengths. By reference to TABLE I, it can be ascertained that the absolute value of the difference between corresponding values of η₁ for Schott BK7 and Schott F2 glasses is quite large, i.e., |η₁₂ -η₁₁ |=0.063435, indicating that the distribution of dioptric powers for lens elements made from these two glasses would be desirably small. Thus, the present invention provides a method for determining the practicality of using two particular glasses, e.g., Schott BK7 glass and Schott F2 glass, in combination for designing a two-glass two-color lens doublet. As an empirical rule, it has been found that where the absolute value of the difference between the values of the primary dispersion coefficients η_(1j) (where j=1, 2) for two optical glasses is greater than about 0.02, i.e., when |Δη_(1j) |=|η₁₂ -η₁₁ | >0.02, the dioptric powers of lens elements made from those two glasses will be small enough so that the lens elements will have sufficiently large radii of curvature to enable a lens doublet of practical configuration to be made.

Also shown in FIG. 8 is a curve showing the relationship of paraxial marginal ray height (conventionally designated PY in the drawing) to wavelength for the lens doublet comprising first and second lens elements made of Schott BK7 and Schott F2 glasses, respectively. Such a curve, hereinafter called a "wavelength scan", for an optical system provides an indication of the quality of axial color correction obtainable with the optical system. The paraxial marginal ray height for an optical system at any particular wavelength is a measure of the so-called secondary spectrum of the system at that wavelength, i.e., a measure of the extent to which light at that particular wavelength is out-of-focus with respect to the common focus for which color-correction of the system is desired. Perfect color correction is achieved for those wavelengths at which the paraxial marginal ray height is zero, i.e., for those wavelengths at which the "wavelength scan" curve crosses the horizontal axis. For wavelengths at which perfect color correction is not achieved, the closeness of the "wavelength scan" curve to the horizontal axis provides a measure of the extent to which the optical system approaches perfect color correction.

For the lens doublet shown in FIG. 8, color correction is obtained at one wavelength in the blue portion of the visible spectrum and at on wavelength in the infrared spectrum. In between these two color-corrected wavelengths, however, the dispersion produced by the lens elements of the doublet is quite pronounced. The "wavelength scan" curve in FIG. 8 indicates that a lens doublet made Schott BK7 and Schott F2 glasses cannot provide a good degree of color correction at wavelengths between the crossings of the horizontal axis, because the curve (i.e., the paraxial marginal ray height) departs rapidly from the horizontal axis even for wavelengths close to the two crossings of the horizontal axis.

INDUSTRIAL APPLICABILITY Two-Glass Lens Doublets

FIG. 9 illustrates a particular example of a three-color cemented lens doublet in which the optical glasses from which the two lens elements of the doublet are made were selected in accordance with the present invention. The lens doublet of FIG. 9 is of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10, and the two glasses used in making the lens elements of the doublet are Schott FK51 and Schott K51 glasses. These glasses were chosen because the ratios of their primary to secondary η-dispersion coefficients are substantially equal, and because the absolute value of the difference between corresponding values of the primary dispersion coefficient η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 9 crosses the horizontal axis at three discrete wavelengths in the visible spectrum, and is close to the horizontal axis throughout the entire wavelength band extending over the visible spectrum. Thus, a lens doublet made from this particular pair of optical glasses can provide very effective (if not perfect) color correction over substantially the entire visible spectrum.

The design form of the lens doublet shown in FIG. 9 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                              Material                                 ______________________________________                                         1       .421 f  .016 f    1.48794                                                                              84.07 Schott FK51                              2      -.215 f  .009 f    1.50720                                                                              59.34 Schott K51                               3      7.034 f                                                                 ______________________________________                                    

where the various lens element surfaces of the doublet system are numbered, in accordance with conventional practice, from left to right. The "Radius" listed for each surface is the radius of curvature of the surface expressed as a multiple of the focal length f of the lens system at a relative aperture of f/10. In accordance with convention, the radius of curvature of an optical surface is said to be positive if the center of curvature of the surface lies to the right of the surface, and negative if the center of curvature of the surface lies to the left of the surface. The "Thickness" listed for a particular surface is the thickness of the lens element bounded on the left by the indicated surface, where the thickness is measured along the optical axis of the lens system. The thickness of each lens element is expressed as a multiple of the focal length f of the lens system. N_(e) is the refractive index of the lens element bounded on the left by the indicated surface, where the value of the refractive index is given for the mercury e line, i.e., for a wavelength of λ₀ =0.54606 μm. V_(e) is the Abbe number for the same lens element at the same base wavelength. "Material" refers to the type of optical material from which the lens element is made, where the type of optical material is identified by manufacturer and by manufacturer's catalog number.

FIG. 10 illustrates a particular example of a four-color cemented lens doublet in which the optical glasses from which the two lens elements of the doublet are made were selected in accordance with the present invention. The lens doublet of FIG. 10 is of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10. The glasses used in making the two lens elements of the doublet of FIG. 10 are Corning France 486817 and Ohara LAK14 glasses, which were chosen because their ratios (η₁ /η₂) are substantially equal in the extended spectral region from 0.365 μm to 1.014 μm, and because the difference between corresponding values of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 10 crosses the horizontal axis at four discrete wavelengths in the visible and near infrared spectrum, and is close to the horizontal axis throughout this spectral range. Thus, a lens doublet made from this particular pair of optical glasses can provide very effective color correction over a wide spectral range bridging the visible and infrared wavelengths, especially in the range extending from the blue wavelengths through the near infrared wavelengths beyond 1.0 μm.

The design form of the lens doublet shown in FIG. 10 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1       .257 f  .016 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             2      -.372 f  .009 f    1.69979                                                                              55.31                                                                               Ohara LAK14                               3      2.022 f                                                                 ______________________________________                                    

where the specified design parameters are as defined above in connection with FIG. 9.

FIG. 11 illustrates a particular example of a four-color air-spaced lens doublet in which the two lens elements are made of the same types of optical glasses as for the cemented lens doublet shown in FIG. 10. The lens doublet of FIG. 11 is likewise of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10. The wavelength scan curve shown in FIG. 11 crosses the horizontal axis at four discrete wavelengths in the visible and near infrared spectrum, and is close to the horizontal axis throughout the same spectral range for which the lens doublet of FIG. 10 provides effective color correction. Thus, the degree of color correction obtainable using the glasses of the cemented lens doublet shown in FIG. 10 is retained when an air space is provided between the lens elements, which implies that the quality of axial (i.e., longitudinal) color correction obtainable with a two-glass lens doublet is determined by the choice of glasses used for making the lens elements, and is independent of the design form of the doublet. The quality of correction for lateral chromatic abberation, spherochromatism and the remaining aberrations, however, depends upon the design form of the lens doublet.

The design form of the lens doublet shown in FIG. 11 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1       .598 f  .016 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             2      -.207 f   .0005 f             Air                                       3      -.221 f  .009 f    1.69979                                                                              55.31                                                                               Ohara LAK14                               4      -.699 f                                                                 ______________________________________                                    

FIG. 12 illustrates a particular example of a five-color cemented lens doublet in which the optical glasses from which the two lens elements of the doublet are made were selected in accordance with the present invention. The lens system of FIG. 12 is of unit focal length at λ₀ =0.54607 μm for a relative aperture of f/10. The glasses used in making the two lens elements of the cemented doublet of FIG. 12 are Schott PK51 and Hoya LAC7 glasses, which were chosen because the ratios of their primary to secondary η-dispersion coefficients are substantially equal, and because the absolute value of the difference between corresponding values of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 12 crosses the horizontal axis at five discrete wavelengths in the spectral range extending from the ultraviolet to the near infrared wavelengths. Furthermore, the wavelength scan curve is extremely close to the horizontal axis throughout the spectral range extending from the green wavelengths to about 0.8 μm in the near infrared region. Thus, a lens doublet made from this particular pair of optical glasses can provide very effective axial color correction throughout this region of the optical spectrum.

The design form of the lens doublet shown in FIG. 12 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                              Material                                 ______________________________________                                         1       .753 f  .020 f    1.53019                                                                              76.57 Schott PK51                              2      -.151 f   .0004 f              Air                                      3      -.156 f  .010 f    1.65426                                                                              58.15 Hoya LAC7                                4      -.680 f                                                                 ______________________________________                                    

Two-Glass Lens Triplets

FIG. 13 illustrates an example of a four-color cemented two-glass lens triplet in which the optical glasses from which the three lens elements of the triplet are made were selected in accordance with the present invention. The lens triplet as shown in FIG. 13 has a focal length that is 40 times the focal length of the lens doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10. The first and third lens elements of the lens triplet of FIG. 13 are made of Corning-France 486817 glass, and the second (i.e., middle) lens element is made of Corning-France 589612 glass. These glass types were chosen because the ratios of their primary to secondary η-dispersion coefficients are substantially equal, and because the absolute value of the difference between corresponding values of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 13 crosses the horizontal axis at four discrete wavelengths in the spectral range extending from the blue to the near infrared wavelengths, and is close to the horizontal axis throughout this spectral range. Thus, a lens triplet made from this particular pair of optical glasses can provide very effective axial color correction throughout this region of the optical spectrum.

It is convenient to specify the design form of the lens triplet of FIG. 13 in terms of unit focal length for a relative aperture of f/10, as was done above for the lens doublets of FIGS. 8-12. It is noted, however, that the depiction in FIG. 13 shows a lens triplet whose focal length is 40 times that of the lens doublets shown in FIGS. 8-12. In terms of the focal length f, the design form of the lens triplet of FIG. 13 is specified in the tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1        .368 f .015 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             2      -2.029 f .009 f    1.59129                                                                              60.94                                                                               Corn-F 589612                             3        .212 f .013 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             4      -2.324 f                                                                ______________________________________                                    

FIG. 14 illustrates a particular example of a four-color two-glass lens triplet in which the three lens elements are made of the same types of optical glasses as for the cemented lens triplet shown in FIG. 13, with the first and second lens elements being cemented together, and with an air gap being provided between the second and third lens elements. The focal length of the lens triplet shown in FIG. 14 is the same as for the lens triplet shown in FIG. 13 for the same relative aperture of f/10.

The wavelength scan curve shown in FIG. 14 likewise crosses the horizontal axis at four discrete wavelengths in the spectral range extending from the blue to the near infrared wavelengths, and is close to the horizontal axis throughout this spectral range. Thus, the degree of axial color correction obtainable using the pair of glasses of the cemented lens triplet shown in FIG. 13 is seen to be retained when an air space is provided between two of the lens elements. The wavelength scan thus indicates that, for a given pair of optical materials, the quality of axial color correction obtainable with a two-glass lens triplet is determined by the choice of glasses used to make the lens elements of the triplet, and is independent of the design form of the triplet. Of course, the quality of correction for lateral color aberration, spherochromatism and the higher-order monochromatic aberrations is dependent upon the design form of the lens triplet.

In terms of the focal length, the design form of the lens triplet of FIG. 14 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1        .419 f .015 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             2       -.449 f .009 f    1.59129                                                                              60.94                                                                               Corn-F 589612                             3        .342 f .001 f               Air                                       4        .360 f .013 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             5      -1.101 f                                                                ______________________________________                                    

Two-Glass Lens Quadruplets

FIG. 15 illustrates a particular example of a four-color cemented two-glass lens quadruplet having a focal length that is 40 times the focal length of the lens doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/10. The optical glasses from which the four lens elements of the quadruplet shown in FIG. 15 are made were selected in accordance with the present invention. The first and third lens elements are made of Corning-France 486817 glass, and the second and fourth lens elements are made of Corning-France 589612 glass. These glasses were chosen because the ratios of their primary to secondary η-dispersion coefficients are substantially equal, and because the absolute value of the difference between corresponding values of η₁ for the two glasses is large.

The wavelength scan curve shown in FIG. 15 crosses the horizontal axis at four discrete wavelengths in the spectral range extending from the blue to the near infrared region of the optical spectrum. The wavelength scan curve is close to the horizontal axis throughout this spectral range, which indicates that a lens quadruplet made from this particular pair of glasses can provide effective axial color correction throughout this spectral range.

In terms of the focal length, the design form of the lens quadruplet of FIG. 15 is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness    N.sub.e                                                                              V.sub.e                                                                              Material                                  ______________________________________                                         1       .437 f   .015 f  1.48746                                                                              81.40 Corn-F 486817                             2      5.098 f   .009 f  1.59129                                                                              60.94 Corn-F 589612                             3       .285 f   .015 f  1.48746                                                                              81.40 Corn-F 486817                             4      -.363 f   .009 f  1.59129                                                                              60.94 Corn-F 589612                             5      -1.147 f                                                                ______________________________________                                    

FIG. 16 illustrates a particular example of a four-color two-glass lens quadruplet in which the four lens elements are made of the same types of optical glasses as for the cemented lens quadruplet shown in FIG. 15, with the first and second lens elements being cemented together, the third and fourth lens elements being cemented together, and an air gap being provided between the second and third lens elements. The focal length of the lens quadruplet shown in FIG. 16 is the same as for the lens quadruplet shown in FIG. 15 for the same relative aperture of f/10.

The wavelength scan curve shown in FIG. 16 likewise crosses the horizontal axis at four discrete wavelengths in the spectral range extending from the blue to the near infrared region of the optical spectrum. The wavelength scan curve of FIG. 16 is likewise close to the horizontal axis throughout this spectral range. As can be seen by comparing the wavelength scan of FIG. 16 with the wavelength scan of FIG. 15, the degree of axial color correction obtainable using the pair of glasses of the cemented lens quadruplet of FIG. 15 is retained when an air space is provided between two of the lens elements.

In terms of the focal length, the design form of the lens quadruplet of FIG. 16 is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness    N.sub.e                                                                              V.sub.e                                                                              Material                                  ______________________________________                                         1       .948 f   .015 f  1.48746                                                                              81.40 Corn-F 486817                             2      -.375 f   .009 f  1.59129                                                                              60.94 Corn-F 589612                             3      1.280 f   .001 f              Air                                       4       .499 f   .009 f  1.59129                                                                              60.94 Corn-F 589612                             5       .274 f   .015 f  1.48746                                                                              81.40 Corn-F 486817                             6      -1.150 f                                                                ______________________________________                                    

FIG. 17 illustrates a particular example of a four-color two-glass lens quadruplet in which the four lens elements are made of the same types of optical glasses as for the lens quadruplets shown in FIGS. 15 and 16, except that the first and third lens elements are made of Corning-France 486817 glass and the second and fourth lens elements are made of Corning-France 589612 glass. The focal length of the lens quadruplet shown in FIG. 17 is the same as for the lens quadruplets shown in FIGS. 15 and 16 for the same relative aperture of f/10. As with the lens quadruplet shown in FIG. 16, the first and second lens elements of the lens quadruplet shown in FIG. 17 are cemented together, the third and fourth lens elements are cemented together, and an air gap is provided between the second and third lens elements.

The wavelength scan curve shown in FIG. 17 likewise crosses the horizontal axis at four discrete wavelengths in the spectral range extending from the blue to the near infrared region of the optical spectrum. The wavelength scan curve of FIG. 16 is likewise close to the horizontal axis throughout this spectral range. When compared with the wavelength scans shown in FIGS. 15 and 16, the wavelengh scan shown in FIG. 17 further indicates that the quality of axial color correction obtainable with a lens multiplet is determined by the choice of glasses used for designing the multiplet, and is independent of the design form of the multiplet.

In terms of the focal length, the design form of the lens quadruplet of FIG. 17 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius   Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1       .823 f  .015 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             2      -.400 f  .009 f    1.59129                                                                              60.94                                                                               Corn-F 589612                             3       1.069 f .001 f               Air                                       4       .645 f  .015 f    1.48746                                                                              81.40                                                                               Corn-F 486817                             5      -.337 f  .009 f    1.59129                                                                              60.94                                                                               Corn-F 589612                             6      -.758 f                                                                 ______________________________________                                    

Two-Glass Collimators

FIG. 18 shows a four-color collimator comprising a two-glass lens quadruplet, which can be used for precise collimation of a beam of light having a spectral range from blue to near infrared. The collimator of FIG. 18 comprises a lens quadruplet wherein the consecutive lens elements are all air-spaced. The first and fourth lens elements of the collimator are made of Hoya FCD10 glass, and the second and third lens elements are made of Ohara LAK14 glass.

The wavelength scan curve shown in FIG. 18 crosses the horizontal axis at four discrete wavelengths, and is close to the horizontal axis throughout a wavelength band that bridges the visible and near infrared spectral regions. In fact, the wavelength scan curve of FIG. 18 substantially coincides with the horizontal axis throughout the near infrared spectral range, which illustrates the suitability of this particular combination of optical glasses for correcting paraxial chromatic aberration in the near infrared spectral range.

The collimator shown in FIG. 18, in addition to being substantially color-corrected in the near infrared spectral range, is also well-corrected for the various monochromatic aberrations. In terms of the focal length for a relative aperture of f/10, the design form of the collimator of FIG. 18 is specified in detail in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness     N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1       1.244 f  .015 f   1.45770                                                                              90.32                                                                               Hoya FCD10                                2      -.381 f   .010 f              Air                                       3      -.381 f   .009 f   1.69979                                                                              55.31                                                                               Ohara LAK14                               4      -.542 f   .002 f              Air                                       5      -.539 f   .009 f   1.69979                                                                              55.31                                                                               Ohara LAK14                               6      -3.845 f   .0005 f            Air                                       7       .715 f   .015 f   1.45770                                                                              90.32                                                                               Hoya FCD10                                8      -1.025 f                                                                ______________________________________                                    

FIG. 19 shows a five-color collimator comprising a two-glass lens quadruplet. The first and fourth lens elements of the collimator of FIG. 19 are made of Schott PK51 glass, and the second and third lens elements are made of Hoya LAC7 glass. These two optical glasses were selected because of the substantial collinearity of their respective glass points with the origin of the coordinate system in four-dimensional η-dispersion coefficient space, and also because the absolute value of the difference between their respective primary dispersion coefficients is large.

The wavelength scan curve shown in FIG. 19 crosses the horizontal axis at five discrete wavelengths in the spectral range extending from about 0.38 μm to about 0.80 μm. These five wavelengths can therefore be precisely focussed using the collimator of FIG. 19. The other wavelengths in this spectral range can be focussed with a degree of precision indicated by the closeness with which the wavelength scan curve approaches the horizontal axis.

The collimator shown in FIG. 19 is well-corrected for the various monochromatic aberrations, especially spherochromatism. In terms of the focal length for a relative aperture of f/10, the design form of the collimator of FIG. 19 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -1.314 f  .018 f    1.53019                                                                              76.57                                                                               Schott PK51                              2      -.295 f   .025 f               Air                                      3      -.269 f   .010 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                4      -1.795 f  .003 f               Air                                      5      .469 f    .010 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                6      .233 f    .002 f               Air                                      7      .232 f    .015 f    1.53019                                                                              76.57                                                                               Schott PK51                              8      -.697 f                                                                 ______________________________________                                    

FIG. 20 shows a five-color collimator comprising a lens quadruplet having a focal length that is 40 times the focal length of the doublets shown in FIGS. 8-12 at λ₀ =0.54607 μm for a relative aperture of f/8. The first and fourth lens elements of the collimator of FIG. 20 are made of calcium fluoride crystal, and the second and third lens elements are made of Hoya LAC7 glass. Use of calcium fluoride crystal as one of the optical materials enables a larger aperture (i.e., a lower f-number) to be used than would be possible if both optical materials were glasses. With the collimator of FIG. 20, a light beam of large diameter can be collimated without significant error due to spherochromatism. The wavelength scan curve shown in FIG. 20 crosses the horizontal axis at five wavelengths, and therefore indicates that precise collimation can be obtained at five wavelengths, in the spectral range from ultraviolet to near infrared. The closeness of the wavelength scan curve to the horizontal axis throughout this spectral range indicates that a high degree of collimation can be obtained for the other wavelengths in this range.

In terms of the focal length for a relative aperture of f/8, the design form of the collimator of FIG. 20 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -.668 f   .018 f    1.43500                                                                              95.06                                                                               Calcium                                                                        Fluoride                                 2      -.268 f   .032 f               Air                                      3      -.250 f   .010 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                4      -.565 f   .001 f               Air                                      5      .455 f    .010 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                6      .384 f    .001 f               Air                                      7      .424 f    .015 f    1.43500                                                                              95.06                                                                               Calcium                                                                        Fluoride                                 8      -.635 f                                                                 ______________________________________                                    

Two-Glass Mangin Mirrors

FIG. 21 illustrates a particular example of a cemented Mangin doublet mirror, which is color-corrected at five wavelengths. The Mangin doublet mirror of FIG. 21 has unit focal length at λ₀ =0.54607 μm for relative aperture of f/2, and comprises a first lens element made of Hoya FCD10 glass and a second lens element made of Hoya FC3 glass. The outward facing surface of the second lens element is provided with a reflective coating, which could comprise a conventional reflecting material such as, e.g., silver or aluminum.

The Hoya FCD10 and Hoya FC3 glasses, used in combination, provide color correction at five wavelengths. The design form of the particular Mangin doublet mirror shown in FIG. 21 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -.958 f   .036 f    1.45770                                                                              90.32                                                                               Hoya FCD10                               2      4.200 f   .057 f    1.46619                                                                              65.49                                                                               Hoya FC3                                 3      -1.572 f  -.057 f   1.46619                                                                              65.49                                                                               Hoya FC3                                 4      4.200 f   -.036 f   1.45770                                                                              90.32                                                                               Hoya FCD10                               5      -.958 f                                                                 ______________________________________                                    

where the negative values of thickness listed for the lens element after the third surface, as numbered in accordance with conventional practice from left to right in the drawing, indicate that reflection occurs at the third surface. Thus, the fourth surface is the same as the second surface, and the fifth surface is the same as the first surface. Light at a color-corrected wavelength passing consecutively through the first and second surfaces is reflected from the third surface back consecutively through the second (i.e., fourth) and first (i.e., fifth) surfaces to the focal plane of the Mangin doublet mirror.

The wavelength scan for the Mangin doublet mirror shown in FIG. 21 indicates that precise color correction occurs at five discrete wavelengths in the optical spectrum, and that a high degree of axial color correction is achieved over a band of wavelengths extending from the far ultraviolet to the near infrared regions of the spectrum. In general, in accordance with the present invention, dispersive effects can be substantially eliminated from Mangin mirrors by the proper choice of optical materials from which the lens elements are made.

FIG. 22 illustrates another example of a cemented Mangin doublet mirror, which provides substantially complete axial color correction from the blue to the near infrared regions of the spectrum. The Mangin doublet mirror of FIG. 22 has unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, and comprises a first lens element made of Hoya BAF11 glass and a second lens element made of Hoya FL6 glass. The outward facing surface of the second lens element is provided with a reflective coating. The design form of the Mangin doublet mirror shown in FIG. 22 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -1.897 f  .025 f    1.67000                                                                              47.97                                                                               Hoya BAF11                               2      3.607 f   .040 f    1.57047                                                                              42.57                                                                               Hoya FL6                                 3      -1.825 f  -.040 f   1.57047                                                                              42.57                                                                               Hoya FL6                                 4      3.607 f   -.025 f   1.67000                                                                              47.97                                                                               Hoya BAF11                               5      -1.897 f                                                                ______________________________________                                    

The wavelength scan for the Mangin doublet mirror of FIG. 22 shows that precise axial color correction (i.e., crossing of the horizontal axis) occurs at four discrete wavelengths, and that a high degree of axial color correction (i.e., only insignificant paraxial marginal ray height) occurs throughout the spectral range extending from the blue to beyond 0.8 μm in the near infrared regions of the spectrum.

FIG. 23 illustrates an example of a cemented Mangin doublet mirror, which provides substantially complete color correction from the blue to well into the near infrared regions of the optical spectrum. The Mangin doublet mirror of FIG. 23 has unit focal length at λ₀ =0.54607 μm for relative aperture of f/2, and comprises a first lens element made of Corning-France 486817 glass and a second lens element made of Corning-France 697554 glass. The outward facing surface of the second lens element is provided with a reflective coating. The design form of the Mangin doublet mirror shown in FIG. 23 is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness     N.sub.e                                                                              V.sub.e                                                                             Material                                  ______________________________________                                         1      -.753 f   .042 f   1.48746                                                                              81.40                                                                               Corn-F 486817                             2      -7.400 f  .066 f   1.70000                                                                              55.18                                                                               Corn-F 697554                             3      -1.536 f  -.066 f  1.70000                                                                              55.18                                                                               Corn-F 697554                             4      -7.400 f  -.042 f  1.48746                                                                              81.40                                                                               Corn-F 486817                             5      -.753 f                                                                 ______________________________________                                    

The wavelength scan for the Mangin doublet mirror of FIG. 23 shows that precise axial color correction occurs at four discrete wavelengths, and that a high degree of axial color correction occurs throughout the spectral range extending from the blue to beyond 1.0 μm in the near infrared region of the spectrum.

FIG. 24 illustrates a particular example of an air-spaced Mangin doublet mirror, which is color-corrected at five wavelengths. The Mangin doublet mirror of FIG. 24 has unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, and comprises a first lens element made of Schott PK51 glass and a second lens element made of Hoya LAC7 glass. The two lens elements are separated by an air gap, and the outward facing surface of the second lens element is provided with a reflective coating. The design form of the Mangin doublet mirror shown in FIG. 24 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -.775 f    .040 f   1.53019                                                                              76.57                                                                               Schott PK51                              2      3.799 f   .007 f               Air                                      3      4.568 f   .061 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                4      -1.501 f  -.061 f   1.65426                                                                              58.15                                                                               Hoya LAC7                                5      4.568 f   -.007 f              Air                                      6      3.799 f   -.040 f   1.53019                                                                              76.57                                                                               Schott PK51                              7      -.775 f                                                                 ______________________________________                                    

The wavelength scan for the Mangin doublet mirror of FIG. 24 indicates that precise color correction occurs at five discrete wavelengths, and that a high degree of axial color correction occurs throughout the spectral range extending from the violet to the near infrared regions of the opical spectrum.

FIG. 25 illustrates an example of an air-spaced Mangin doublet mirror, which provides substantially complete axial color correction throughout the visible and into the near infrared portions of the optical spectrum. The Mangin doublet mirror of FIG. 25 has unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, and comprises a first lens element made of Hoya FCD10 glass and a second lens element made of Ohara PSK1 glass. The two lens elements are separated by an air gap, and the outward facing surface of the second lens element is provided with a reflective coating. The design form of the Mangin doublet mirror shown in FIG. 25 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -.736 f   .036 f    1.45770                                                                              90.32                                                                               Hoya FCD10                               2      8.012 f   .004 f               Air                                      3      8.273 f   .057 f    1.54979                                                                              62.60                                                                               Ohara PSK1                               4      -1.476 f  -.057 f   1.54979                                                                              62.60                                                                               Ohara PSK1                               5      8.273 f   -.004 f              Air                                      6      8.012 f   -.036 f   1.45770                                                                              90.32                                                                               Hoya FCD10                               7      -.736 f                                                                 ______________________________________                                    

The wavelength scan curve shown in FIG. 25 substantially coincides with the horizontal axis throughout the visible portion of the optical spectrum, although a larger scale plot would show that the crossings of the horizontal axis actually occur at five discrete wavelengths. Technically, therefore, the Mangin doublet mirror of FIG. 25 is color-corrected at only five wavelengths. However, for many practical applications, this Mangin doublet mirror can be considered to be substantially color-corrected throughout the visible wavelength band.

FIG. 26 illustrates an example of an air-spaced Mangin doublet mirror, which provides substantially complete axial color correction from the green region of the visible spectrum to well into the infrared wavelengths. The Mangin doublet mirror of FIG. 26 has unit focal length at λ₀ =0.54607 μm for a relative aperture of f/2, and comprises a first lens element made of Hoya FCD10 glass and a second lens element made of Hoya LAC14 glass. The two lens elements are separated by an air gap, and the outward facing surface of the second lens element is provided with a reflective coating. The design form of the Mangin doublet mirror shown in FIG. 26 is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      -.661 f   .029 f    1.45770                                                                              90.32                                                                               Hoya FCD10                               2      -2.254 f   .010 f              Air                                      3      -3.628 f   .054 f   1.69980                                                                              55.25                                                                               Hoya LAC14                               4      -1.519 f  -.054 f   1.69980                                                                              55.25                                                                               Hoya LAC14                               5      -3.628 f  -.010 f              Air                                      6      -2.254 f  -.029 f   1.45770                                                                               9.32                                                                               Hoya FCD10                               7       -.661 f                                                                ______________________________________                                    

The wavelength scan curve shown in FIG. 26 substantially coincides with the horizontal axis from the green wavelength region to beyond 1.0 μm in the near infrared wavelength region of the spectrum. A larger scale plot would show that crossings of the horizontal axis actually occur at only four discrete wavelengths. Technically, therefore, the Mangin doublet mirror of FIG. 26 is color-corrected at only four wavelengths. However, for many practical applications, particularly applications in the near infrared region of the spectrum, the Mangin doublet mirror of FIG. 26 can be considered to be substantially color-corrected throughout the wavelength band of interest.

Two-Glass Petzval Lens Systems

The designation "Petzval lens system" is generally applied to a lens system comprising two widely separated lens groups, each of which has a net positive dioptric power, followed by a negative lens element or a lens group having a net negative dioptric power. The positive lens groups in combination provide good resolution at high aperture for narrow to moderate fields of view. The negative lens element or lens group functions as a field flattener. A lens group is said to have positive dioptric power when it causes light rays passing through it to converge, and to have negative dioptric power when it causes light rays passing through it to diverge. Often, each lens group of a Petzval lens system is a doublet. However, for a particular application, either one (or all) of the lens groups of a Petzval lens system could be a singlet, a triplet or a high-number multiplet.

The spacing between the two positive lens groups of a Petzval lens system is large in comparison with the thicknesses of the individual lens elements comprising each lens group. Within a lens group, the individual lens elements could be cemented or air-spaced. It is usual for a negative lens element or lens group functioning as a field flattener to be positioned immediately in front of the image plane of the system to correct for field curvature. A Petzval lens system that includes a field flattener is called, appropriately, a flat-field Petzval lens system.

For a given optical design, aberrations scale linearly with the physical size of the system. Thus, a lens system that is reasonably well-corrected for a relatively short focal length (say, less than 10 mm) might have very severe aberration problems at a focal length of more than 100 mm. In the prior art, Petzval lens systems were typically used only in short-focus applications, e.g., in projectors for 16 mm and 8 mm motion-picture films. For very long-focus applications (e.g., more than 200 mm), Petzval-type lens systems were not generally used because of the difficulty in designing high-aperature long-focus Petzval lens systems that are well-corrected.

The lens group of Petzval lens systems of the prior art were typically "achromats", i.e., doublets color-corrected at only two wavelengths. Furthermore, Petzval lens systems of the prior art were typically used only in relatively narrow field-angle applications. An example of a short-focus narrow field-angle Petzval lens system is described in U.S. Pat. No. 3,255,664.

FIG. 27 shows a two-glass flat-field lens system of the Petzval type made of the same two types of optical glasses as used for the lens doublet shown in FIG. 12, i.e., Schott PK51 and Hoya LAC7 glasses. The last lens element in the system, which is a biconcave lens made of Hoya LAC7 glass, functions as a field flattener. The Petzval lens system of FIG. 27 is color-corrected at five wavelengths in the optical spectrum from 0.4 μm to 0.7 μm, and is also well-corrected for monochromatic aberrations in this spectral range.

The design form of the Petzval lens system of FIG. 27, in terms of the focal length for a relative aperture of f/3.5, is specified in tabular format as follows:

    ______________________________________                                         Surface                                                                        No.    Radius    Thickness N.sub.e                                                                              V.sub.e                                                                             Material                                 ______________________________________                                         1      .342 f    .031 f    1.53019                                                                              76.57                                                                               Schott PK51                              2      -.497 f   .006 f               Air                                      3      -1.190 f  .020 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                4      1.465 f   .015 f               Air                                      5      -.299 f   .020 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                6      .522 f    .001 f               Air                                      7      .350 f    .028 f    1.53019                                                                              76.57                                                                               Schott PK51                              8      -.438 f   .274 f               Air                                      9      .508 f    .023 f    1.53019                                                                              76.57                                                                               Schott PK51                              10     -.593 f   .016 f               Air                                      11     -.225 f   .018 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                12     1.483 f   .014 f               Air                                      13     .417 f    .025 f    1.53019                                                                              76.57                                                                               Schott PK51                              14     -.356 f   .268 f               Air                                      15     -.241 f   .013 f    1.65426                                                                              58.15                                                                               Hoya LAC7                                16     2.412 f                                                                 ______________________________________                                    

The wavelength scan for the Petzval lens system of FIG. 27 shows three crossings of the horizontal axis in the visible spectral range, as well as single crossings in the ultraviolet and infrared portions of the spectrum. For the visible wavelengths from the blue region into the red region, the paraxial marginal ray height is insignificant, indicating that very substantial axial color correction occurs throughout that portion of the visible spectrum. The design form of this Petzval lens system also provides superb correction of the monochromatic aberrations, and the chromatic variation of spherical aberration and coma.

The Petzval lens system of FIG. 27 is noteworthy for having a long focal length (f=600 mm) at a relative aperture of f/3.5, while providing a field of view of 12°. This particular Petzval lens system is without precedent in enabling diffraction limited performance to be provided at such a large aperture and wide field of view.

FIG. 28 shows a two-glass flat-field lens system of the Petzval type made of the same two types of optical glasses as used for the lens doublets shown in FIGS. 10 and 11, i.e., Corning-France 486817 and Ohara LAK14 glasses. The Petzval lens system of FIG. 28 is color-corrected at four wavelengths, two in the visible region and two in the infrared region. This system is also well-corrected for spherochromatism and the monochromatic aberrations.

The design form of the Petzval lens system of FIG. 28, in terms of the focal length for a relative aperture of f/3.5, is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness    N.sub.e                                                                               V.sub.e                                                                             Material                                  ______________________________________                                         1       .305 f   .020 f   1.48746                                                                              81.40                                                                               Corn-F 486817                             2      -.900 f   .005 f              Air                                       3      -.337 f   .013 f   1.69979                                                                              55.31                                                                               Ohara LAK14                               4       .398 f   .001 f              Air                                       5       .192 f   .018 f   1.48746                                                                              81.40                                                                               Corn-F 486817                             6      -.692 f   .166 f              Air                                       7       .441 f   .016 f   1.48746                                                                              81.40                                                                               Corn-F 486817                             8      -.385 f   .010 f              Air                                       9      -.146 f   .011 f   1.69979                                                                              55.31                                                                               Ohara LAK14                               10     -.552 f   .009 f              Air                                       11      .393 f   .060 f   1.48746                                                                              81.40                                                                               Corn-F 486817                             12     -.240 f   .187 f              Air                                       13     -.162 f   .014 f   1.69979                                                                              55.31                                                                               Ohara LAK14                               14      .789 f                                                                 ______________________________________                                    

The wavelength scan for the Petzval lens system of FIG. 28 shows very substantial (if not perfect) color-correction throughout the spectral range extending from the blue wavelength region to beyond 1.0 μm in the near infrared region. It is remarkable that such color correction can be obtained for a Petzval lens system using only two different optical glasses. The Petzval lens system of FIG. 28 has a long focal length (f=380 mm) at a relative aperture of f/3.5, while providing a field of view of approximately 12°. The design form of this Petzval lens system provides diffraction limited performance throughout the spectral range for which it is color-corrected.

FIG. 29 shows another long-focus flat-field Petzval lens system whose lens elements are made of only two different types of optical glasses, viz., Schott FK51 and Schott LAKN14 glasses. This Petzval lens system is color-corrected at three wavelengths, two in the visible region and one in the infrared region. The system is also well-corrected for spherochromatism and the monochromatic aberrations, and provides diffraction limited performance thoughout the spectral range for which it is color-corrected.

The design form of the Petzval lens system of FIG. 29, in terms of the focal length for a relative aperture of f/2 (i.e., an extremely wide aperture), is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness    N.sub.e                                                                              V.sub.e                                                                              Material                                  ______________________________________                                         1      2.066 f   .094 f  1.48794                                                                              84.07 Schott FK51                               2      -1.933 f  .156 f              Air                                       3      4.568 f   .087 f  1.48794                                                                              84.07 Schott FK51                               4      -1.026 f  .004 f              Air                                       5      -.991 f   .037 f  1.69980                                                                              55.19 Schott LAKN14                             6      24.483 f  .033 f              Air                                       7      -1.339 f  .037 f  1.69980                                                                              55.19 Schott LAKN14                             8      63.427 f  .001 f              Air                                       9      .850 f    .087 f  1.48794                                                                              84.07 Schott FK51                               10     -5.545 f  .873 f              Air                                       11     .422 f    .050 f  1.48794                                                                              84.07 Schott FK51                               12     -.555 f   .010 f              Air                                       13     -.473 f   .025 f  1.69980                                                                              55.19 Schott LAKN14                             14     -5.724 f  .032 f              Air                                       15     -2.021 f  .025 f  1.69980                                                                              55.19 Schott LAKN14                             16     .969 f    .001 f              Air                                       17     .340 f    .050 f  1.48794                                                                              84.07 Schott FK51                               18     -.798 f   .249 f              Air                                       19     -.252 f   .025 f  1.69980     Schott LAKN14                             20     5.189 f                                                                 ______________________________________                                    

The wavelength scan for the Petzval lens system of FIG. 29 shows that precise axial color-correction occurs at the wavelengths 0.527 μm, 0.640 μm, and 0.778 μm. In the range from 0.527 μm to 0.778 μm, the degree of color-correction obtainable is very high. The Petzval lens system of FIG. 29 has a focal length of 1000 mm, and a field angle of 4.2°.

Two-Glass Prism Systems

FIG. 30 illustrates a two-glass cemented doublet prism system having a deviation angle of 6° for which very substantial color correction (i.e., only insignificant dispersion) occurs throughout the visible spectrum. The deviation angle of a prism is defined as the angle through which a ray changes direction after passing through the prism, i.e., the angle between the direction of the ray entering the system and the direction of the ray exiting from the system. The prism system of FIG. 30 comprises a first prism element made of Schott PK51 glass and having an apex angle of α₁ =42.0239°, and a second prism element made of Hoya LAC7 glass and having an apex angle of α₂ =25.5872°. The negative sign for α₂ is a convention indicating that the orientation of the apex of the second prism element is inverted with respect to the orientation of the apex of the first prism element.

Also shown in FIG. 30 is a plot of change in angular deviation (in degrees) versus wavelength (in micrometers) for this particular doublet prism system for a 6° total deviation angle. The curve in FIG. 30 crosses the horizontal axis at five discrete wavelengths, which indicates that chromatic dispersion from a total deviation angle of 6° is precisely zero at those discrete wavelengths. It is particularly noteworthy, however, that the curve in FIG. 30 is very close to the horizontal axis throughout the entire visible region extending well into the infrared region of the spectrum to about 0.8 μm. For those wavelengths at which the total deviation angle differs from 6°, the difference is approximately 0.0001 degree throughout this wavelength band. Thus, for many applications, the prism system of FIG. 30 can be considered to be color-corrected (i.e., as having substantially zero dispersion) for a deviation angle of 6° throughout this rather wide wavelength band, which includes the entire visible region.

The plot of change in angular deviation versus wavelength shown in FIG. 30 describes a "real" ray trace, not a paraxial ray trace as in the foregoing plots for dioptric and catadioptric systems. The plots shown in FIG. 30 and in the following examples of other prism systems were based upon data obtained from ray traces according to Snell's law.

FIG. 31 illustrates a two-glass cemented quintuplet prism system, which provides a high degree of color correction throughout the visible spectrum for a total deviation angle of 30°. The prism system of FIG. 31 comprises a first prism element made of Schott PK51 glass and having an apex angle of α₁ =43.1695°, a second prism element made of Hoya LAC7 glass and having an apex angle of α₂ =43.1695°, a third prism element made of Schott PK51 glass and having an apex angle of α₃ =43.1695°, a fourth prism element made of Hoya LAC7 glass and having an apex angle of α₄ =43.1695°, and a fifth prism element made of Schott PK51 glass and having an apex angle of α₅ =57.5776°. From a manufacturing standpoint, it is of special interest that α₁, α₂, α₃ and α₄ have the same value, where the negative sign given for α₂ and α₄ indicates that the orientations of the second and fourth prism elements are inverted with respect to the orientations of the first, third and fifth prism elements.

A plot of change in angular deviation versus wavelength shown in FIG. 31 indicates that chromatic dispersion from a total deviation angle of 30° is precisely zero at five discrete wavelengths. It is to be noted that a total deviation angle of 30° is a large total deviation angle. The 30° deviation of a beam of light that is produced by the prism system of FIG. 31 is equivalent to the deviation produced by the marginal zone of a lens with a relative aperture of f/1.0. The plot of change in angular deviation versus wavelength in FIG. 31 indicates that the change in total angular deviation from 30° is only about 0.0005 degree for any wavelength in the spectral band that includes the entire visible region and a portion of the near infrared region extending to almost 0.8 μm. The change in total angular deviation from 30° is in fact zero at five discrete wavelengths in this spectral band.

FIG. 32 illustrates a two-glass cemented quintuplet prism system, which is designed using the same types of glasses as the quintuplet prism system of FIG. 31. The only difference between the prism system of FIG. 32 and the prism system of FIG. 31 is that the apex angles of the prism elements of the system shown in FIG. 32 have different values. Thus, for the first prism element α₁ =83.1964°, for the second prism element α₂ =44.9553°, for the third prism element α₃ =40.8576°, for the fourth prism element α₄ =46.9210°, and for the fifth prism element α₅ =37.5019°.

Optical Systems Comprising More Than Two Glasses

The present invention provides the designer with the ability to design optical systems that are color-corrected at three or more wavelengths using only two different optical materials. Thus, as a result of the present invention, complicated multiglass optical systems of the prior art can be replaced by simpler two-glass optical systems that provide comparable, or better, color correction. This ability to simplify complicated optical systems while improving performance is a major advantage inherent in the present invention.

The method of the present invention for selecting more than two optical materials for designing color-corrected optical systems is of general interest in terms of demonstrating the unified theory. Ordinarily, in practical applications, a two-glass optical system is to be preferred to a system having the same capabilities but comprising more than two optical materials. In particular applications, however, where color correction at particular wavelengths is required, optical systems comprising three or more different optical materials might be used to provide optimum color correction for specified operational requirements.

FIG. 33 illustrates a particular example of a three-glass lens multiplet for which the lens elements comprising the multiplet were selected in accordance with the present invention. The lens multiplet of FIG. 33 is a quintuplet having a focal length of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/10, and provides color correction at five wavelengths. In accordance with conventional practice, the five lens elements comprising the lens quintuplet are described consecutively from left to right as shown in drawing. The design form of the lens quintuplet of FIG. 33 is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness    N.sub.e                                                                              V.sub.e                                                                              Material                                  ______________________________________                                         1      .493 f    .015 f  1.49845                                                                              81.22 Ohara FK01                                2      -.660 f   .034 f              Air                                       3      -.538 f   .012 f  1.72538                                                                              34.49 Ohara KZFS8                               4      1.078 f   .012 f  1.49845                                                                              81.22 Ohara FK01                                5      -2.457 f  .001 f              Air                                       6      -5.045 f  .013 f  1.70443                                                                              29.89 Ohara SF15                                7      -.288 f   .013 f  1.72538                                                                              34.49 Ohara KZFS8                               8      -.953 f                                                                 ______________________________________                                    

The wavelength scan for the three-glass multiplet shown in FIG. 33 indicates that a lens system can be designed using Ohara FKO1, Ohara KZFS8 and Ohara FKO1 glasses for bringing five wavelength from the ultraviolet through the visible to the near infrared regions of the spectrum to a common focus. The wavelength scan further indicates that substantially complete axial color correction can be provided by these three glasses in combination for wavelengths in the near infrared region up to at least 1.0 μm.

FIG. 34 illustrates a particular example of a four-glass lens multiplet for which the lens elements comprising the multiplet were selected in accordance with the present invention. The lens multiplet of FIG. 34 is a quintuplet having a focal length of 1000 mm at λ₀ =0.54607 μm for a relative aperture of f/10, and provides color correction at five wavelengths. The design form of the lens quintuplet of FIG. 34 is specified in tabular format as follows:

    ______________________________________                                         Surface          Thick-                                                        No.    Radius    ness    N.sub.e                                                                              V.sub.e                                                                             Material                                   ______________________________________                                         1      .425 f    .015 f  1.53019                                                                              76.57                                                                               Schott PK51                                2      -.749 f   .008 f             Air                                        3      -.369 f   .012 f  1.69669                                                                              51.28                                                                               Schott LAK20                               4       .416 f   .015 f  1.53019                                                                              76.57                                                                               Schott PK51                                5      -.385 f   .001 f             Air                                        6       .546 f   .015 f  1.61669                                                                              44.07                                                                               Schott KZFSN4                              7      -.170 f   .012 f  1.59487                                                                              48.29                                                                               Schott KZFS6                               8       .434 f                                                                 ______________________________________                                    

The wavelength scan for the four-glass multiplet shown in FIG. 34 indicates that a lens system can be designed using Schott PK51, Schott LAK20, Schott KZFSN4 and Schott KZFS6 glasses for bringing five wavelengths from the ultraviolet through the visible to the near infrared regions of the spectrum to a common focus. The wavelength scan further indicates that substantially complete color correction can be provided by these four glasses in combination for wavelengths in the range from about 0.60 μm to 0.85 μm.

CONCLUSION

This invention has been described above in terms of particular optical systems designed using particular glasses and other types of optical materials that are commercially available at the present time. However, other kinds of optical systems, which could be designed using other types of glasses and optical materials, could be made by designers skilled in the art on the basis of the foregoing disclosure in conjunction with the accompanying drawing. Therefore, the novel optical systems described above are to be considered as merely illustrative of the invention. The legal definition of the invention is provided more generally by the following claims and their equivalents.

                                      TABLE I                                      __________________________________________________________________________     LEAST SQUARES FITTED COEFFICIENTS                                                           ORDER                                                                               BASE WVL                                                                              MIN VL                                                                              MAX WVL                                                       2.0  0.5740 0.400                                                                               0.700                                            CAT   NAME  ETA1/ETA2                                                                             INDEX                                                                               ETA1   ETA2   ARCTAN(ETA1/ETA2)                        __________________________________________________________________________     OHARA SK4   -.500727+05                                                                           1.613471                                                                            -.939625-01                                                                           .187652-05                                                                            -89.998856                               OHARA LAKO10                                                                               -.324490+04                                                                           1.726980                                                                            -.102990+00                                                                           .317391-04                                                                            -89.982343                               OHARA LAK11 -.294092+04                                                                           1.659127                                                                            -.962812-01                                                                           .327384-04                                                                            -89.980517                               CHANCE                                                                               691547                                                                               -.192659+04                                                                           1.691914                                                                            -.100937+00                                                                           .523916-04                                                                            -89.970261                               SCHOTT                                                                               SK15  -.148992+04                                                                           1.623767                                                                            -.951161-01                                                                           .638397-04                                                                            -89.961544                               OHARA ZK1   -.113189+04                                                                           1.533817                                                                            -.952528-01                                                                           .841541-04                                                                            -89.949380                               SCHOTT                                                                               SK51  -.882619+03                                                                           1.621646                                                                            -.915519-01                                                                           .103728-03                                                                            -89.935084                               SCHOTT                                                                               BAK5  -.863211+03                                                                           1.557397                                                                            -.941415-01                                                                           .109060-03                                                                            -89.933624                               SCHOTT                                                                               ZK1   -.542770+03                                                                           1.533815                                                                            -.952493-01                                                                           .175487-03                                                                            -89.894439                               OHARA LAK8  -.505624+03                                                                           1.713954                                                                            -.102540+00                                                                           .202799-03                                                                            -89.886683                               CORN-F                                                                               713538                                                                               -.420221+03                                                                           1.713958                                                                            -.102594+00                                                                           .244142-03                                                                            -89.863654                               HOYA  LAC12 -.391017+03                                                                           1.678786                                                                            -.994099-01                                                                           .254234-03                                                                            -89.853470                               OHARA SK15  -.360639+03                                                                           1.623767                                                                            -.949577-01                                                                           .263304-03                                                                            -89.841127                               HOYA  C12   -.346402+03                                                                           1.523719                                                                            -.941556-01                                                                           .271810-03                                                                            -89.834599                               HOYA  C3    -.307862+03                                                                           1.518870                                                                            -.936381-01                                                                           .304156-03                                                                            -89.813891                               OHARA ZK4   -.297025+03                                                                           1.512537                                                                            -.949230-01                                                                           .319579-03                                                                            -89.807101                               SCHOTT                                                                               SK13  -.245242+03                                                                           1.592544                                                                            -.946779-01                                                                           .386059-03                                                                            -89.766372                               CHANCE                                                                               572577                                                                               -.238564+03                                                                           1.572917                                                                            -.956311-01                                                                           .400862-03                                                                            -89.759831                               CORN-F                                                                               651559                                                                               -.207410+03                                                                           1.651968                                                                            -.987661-01                                                                           .476189-03                                                                            -89.723757                               OHARA BAK1  -.203696+03                                                                           1.573218                                                                            -.956013-01                                                                           .469332-03                                                                            -89.718722                               HOYA  BACD13                                                                               -.195377+03                                                                           1.592545                                                                            -.946756-01                                                                           .484579-03                                                                            -89.706745                               CORN-F                                                                               658572                                                                               -.193118+03                                                                           1.658335                                                                            -.965192-01                                                                           .499794-03                                                                            -89.703314                               OHARA K2    -.172343+03                                                                           1.516673                                                                            -972221-01                                                                            .564120-03                                                                            -89.667551                               OHARA BAK5  -.165965+03                                                                           1.557399                                                                            -.941142-01                                                                           .567071-03                                                                            -89.654778                               OHARA KO1   -.146523+03                                                                           1.523717                                                                            -.944011-01                                                                           .644277-03                                                                            -89.608969                               OHARA LAK12 -.138588+03                                                                           1.678787                                                                            -.997704-01                                                                           .719909-03                                                                            -89.586580                               CORN-F                                                                               691548                                                                               -.127310+03                                                                           1.691921                                                                            -.100767+00                                                                           .791505-03                                                                            -89.549962                               SCHOTT                                                                               SK52  -.953341+02                                                                           1.639373                                                                            -.993953-01                                                                           .104260-02                                                                            -89.399022                               OHARA SK10  -.921555+02                                                                           1.623589                                                                            -.966943-01                                                                           .104925-02                                                                            -89.378296                               OHARA LAK13 -.907372+02                                                                           1.694441                                                                            -.103439+00                                                                           .113999-02                                                                            -89.368575                               OHARA LAKO11                                                                               -.897967+02                                                                           1.742014                                                                            -.104780+00                                                                           .116686-02                                                                            -89.361963                               OHARA LAKO13                                                                               -.830518+02                                                                           1.641813                                                                            -.969394-01                                                                           .116722-02                                                                            -89.310151                               SCHOTT                                                                               K10   -.824042+02                                                                           1.502014                                                                            -.978985-01                                                                           .118803-02                                                                            -89.304731                               OHARA SK19  -.787332+02                                                                           1.614193                                                                            -.961291-01                                                                           .122095-02                                                                            -89.272317                               SCHOTT                                                                               SK19  -.783834+02                                                                           1.614195                                                                            -.961830-01                                                                           .122708-02                                                                            -89.269073                               HOYA  BACD10                                                                               -.768258+02                                                                           1.623591                                                                            -.969209-01                                                                           .126157-02                                                                            -89.254251                               SCHOTT                                                                               BAK1  -.751939+02                                                                           1.573219                                                                            -.958644-01                                                                           .127490-02                                                                            -89.238069                               HOYA  BACD1 -.743045+02                                                                           1.611028                                                                            -.974048-01                                                                           .131089-02                                                                            -89.228950                               HOYA  LACL2 -.727112+02                                                                           1.651040                                                                            -.992233-01                                                                           .136462-02                                                                            -89.212060                               HOYA  BACD6 -.716787+02                                                                           1.614541                                                                            -.978361-01                                                                           .136493-02                                                                            -89.200707                               SCHOTT                                                                               LAKN22                                                                               -.715624+02                                                                           1.651973                                                                            -.987132-01                                                                           .137940-02                                                                            -89.199411                               CHANCE                                                                               623569                                                                               -.657657+02                                                                           1.623589                                                                            -.969717-01                                                                           .147450-02                                                                            -89.128858                               SCHOTT                                                                               SK10  -.654909+02                                                                           1.623593                                                                            -.969513-01                                                                           .148038-02                                                                            -89.125200                               HOYA  SBF5  -.591198+02                                                                           1.522013                                                                            -.104595+00                                                                           .176920-02                                                                            -89.030948                               OHARA LASKO1                                                                               -.581632+02                                                                           1.756042                                                                            -.105438+00                                                                           .181279-02                                                                            -89.015014                               SCHOTT                                                                               SK2   -.566100+02                                                                           1.608157                                                                            -.973525-01                                                                           .171971-02                                                                            -88.987988                               CORN-F                                                                               623569                                                                               -.561077+02                                                                           1.623593                                                                            -.969182-01                                                                           .172736-02                                                                            -88.978932                               CHANCE                                                                               569561                                                                               -.560878+02                                                                           1.569560                                                                            -.982912-01                                                                           .175245-02                                                                            -88.978572                               CHANCE                                                                               678552                                                                               -.556141+02                                                                           1.678788                                                                            -.999158-01                                                                           .179659-02                                                                            -88.969873                               SCHOTT                                                                               SK1   -.524111+02                                                                           1.611029                                                                            -.972537-01                                                                           .185559-02                                                                            -88.906936                               CORN-F                                                                               614564                                                                               -.520142+02                                                                           1.614285                                                                            -.977645-01                                                                           .187957-02                                                                            -88.898597                               HOYA  BACD2 -.519914+02                                                                           1.608159                                                                            -.972569-01                                                                           .187064-02                                                                            -88.898108                               HOYA  ADC1  -.515199+02                                                                           1.620722                                                                            -.886488-01                                                                           .172067-02                                                                            -88.888030                               HOYA  BAC4  -.500459+02                                                                           1.569563                                                                            -.984724-01                                                                           .196764-02                                                                            -88.855289                               OHARA BAK6  -.497699+02                                                                           1.575177                                                                            -.976194-01                                                                           .196141-02                                                                            -88.848944                               SCHOTT                                                                               LAKN16                                                                               -.481711+02                                                                           1.734527                                                                            -.106819+00                                                                           .221749-02                                                                            -88.810749                               CHANCE                                                                               691548                                                                               -.474013+02                                                                           1.691911                                                                            -.100650+00                                                                           .212337-02                                                                            -88.791434                               SCHOTT                                                                               SK6   -.472162+02                                                                           1.614537                                                                            -.977687-01                                                                           .207066-02                                                                            -88.786704                               OHARA SK1   -.470107+02                                                                           1.611035                                                                            -.975581-01                                                                           .207523-02                                                                            -88.781403                               SCHOTT                                                                               BAK4  -.466145+02                                                                           1.569563                                                                            -.982860-01                                                                           .210849-02                                                                            -88.771045                               OHARA BAK4  -.465233+02                                                                           1.569563                                                                            -.982470-01                                                                           .211178-02                                                                            -88.768640                               OHARA LAKO4 -.461629+02                                                                           1.651835                                                                            -.982167-01                                                                           .212761-02                                                                            -88.759029                               SCHOTT                                                                               BAK6  -.446430+02                                                                           1.575176                                                                            -.978150-01                                                                           .219105-02                                                                            -88.716792                               OHARA SK6   -.445799+02                                                                           1.614537                                                                            -.978164-01                                                                           .219418-02                                                                            -88.714978                               SCHOTT                                                                               LAKN12                                                                               -.438037+02                                                                           1.678788                                                                            -.998759-01                                                                           .228008-02                                                                            -88.692214                               HOYA  BACD8 -.438015+02                                                                           1.611961                                                                            -.988673-01                                                                           .225717-02                                                                            -88.692147                               HOYA  BAC1  -.418595+02                                                                           1.573217                                                                            -.959622-01                                                                           .229248-02                                                                            -88.631498                               OHARA SK2   -.415061+02                                                                           1.608152                                                                            -.970336-01                                                                           .233782-02                                                                            -88.619845                               CHANCE                                                                               734517                                                                               -.399053+02                                                                           1.734527                                                                            -.106738+00                                                                           .267478-02                                                                            -88.564508                               SCHOTT                                                                               SK8   -.396332+02                                                                           1.611961                                                                            -.985867-01                                                                           .248748-02                                                                            -88.554654                               OHARA KZF5  -.386798+02                                                                           1.522016                                                                            -.104984+00                                                                           .271418-02                                                                            -88.519047                               OHARA LAKO9 -.386447+02                                                                           1.735028                                                                            -.107216+00                                                                           .277441-02                                                                            -88.517697                               OHARA KF3   -.382488+02                                                                           1.515220                                                                            -.100854+00                                                                           .263679-02                                                                            -88.502365                               CORN-F                                                                               569560                                                                               - .377078+02                                                                          1.569531                                                                            -.984073-01                                                                           .260973-02                                                                            -88.480890                               CORN-F                                                                               734514                                                                               -.374738+02                                                                           1.734545                                                                            -.107304+00                                                                           .286344-02                                                                            -88.471407                               OHARA SK8   -.370859+02                                                                           1.611956                                                                            -.985702-01                                                                           .265789-02                                                                            -88.455426                               OHARA ZK5   -.369012+02                                                                           1.534444                                                                            -.992690-01                                                                           .269013-02                                                                            -88.447699                               SCHOTT                                                                               SKN18 -.361546+02                                                                           1.639374                                                                            -.994923-01                                                                           .275186-02                                                                            -88.415658                               HOYA  TAC4  -.359864+02                                                                           1.735038                                                                            -.108040+00                                                                           .300226-02                                                                            -88.408251                               HOYA  BAC6  -.349833+02                                                                           1.575178                                                                            -.978458-01                                                                           .279693-02                                                                            -88.362641                               HOYA  BACD18                                                                               -.343625+02                                                                           1.639374                                                                            -.994040-01                                                                           .289281-02                                                                            -88.333075                               SCHOTT                                                                               LAK28 -.331265+02                                                                           1.745347                                                                            -.108591+00                                                                           .327807-02                                                                            -88.270920                               CORN-F                                                                               639555                                                                               -.330270+02                                                                           1.639336                                                                            -.993028-01                                                                           .300671-02                                                                            -88.265717                               HOYA  BACED4                                                                               -.326817+02                                                                           1.618461                                                                            -.999157-01                                                                           .305724-02                                                                            -88.247398                               CORN-F                                                                               529518                                                                               -.323428+02                                                                           1.529735                                                                            -.106441+00                                                                           .329102-02                                                                            -88.229052                               HOYA  TAF1  -.321194+02                                                                           1.773625                                                                            -.111086+00                                                                           .345853-02                                                                            -88.216740                               SCHOTT                                                                               KF3   -.321153+02                                                                           1.515220                                                                            -.100799+00                                                                           .313867-02                                                                            -88.216505                               HOYA  ZNC5  -.317057+02                                                                           1.534442                                                                            -.995644-01                                                                           .314026-02                                                                            -88.193491                               SCHOTT                                                                               KZF2  -.308127+02                                                                           1.530180                                                                            -.106733+ 00                                                                          .346394-02                                                                            -88.141160                               CHANCE                                                                               639554                                                                               -.303756+02                                                                           1.639373                                                                            -.994654-01                                                                           .327451-02                                                                            -88.114441                               SCHOTT                                                                               SSK4  -.301555+02                                                                           1.618460                                                                            -.999556-01                                                                           .331467-02                                                                            -88.100686                               OHARA SK18  -.298634+02                                                                           1.639372                                                                            -.995302-01                                                                           .333286-02                                                                            -88.082115                               SCHOTT                                                                               ZK5   -.298507+02                                                                           1.534448                                                                            -.996534-01                                                                           .333839-02                                                                            -88.081308                               HOYA  NBF3  -.297358+02                                                                           1.736068                                                                            -.110662+00                                                                           .372151-02                                                                            -88.073895                               CORN-F                                                                               618551                                                                               -.293313+02                                                                           1.618415                                                                            -.999977-01                                                                           .340924-02                                                                            -88.047360                               HOYA  NBF1  -.292015+02                                                                           1.744391                                                                            -.111894+00                                                                           .383180-02                                                                            -88.038677                               HOYA  BACD9 -.290570+02                                                                           1.614853                                                                            -.100019+00                                                                           .344216-02                                                                            -88.028940                               OHARA SSK4  -.289407+02                                                                           1.618457                                                                            -.100125+00                                                                           .345966-02                                                                            -88.021023                               OHARA SK9   -.284730+02                                                                           1.614855                                                                            -.100310+00                                                                           .352300-02                                                                            -87.988535                               OHARA KZF2  -.283086+02                                                                           1.530181                                                                            -.106660+00                                                                           .376777-02                                                                            -87.976865                               SCHOTT                                                                               SK9   -.279606+02                                                                           1.614854                                                                            -.998924-01                                                                           .357262-02                                                                            -87.951707                               HOYA  LACL4 -.276702+02                                                                           1.670540                                                                            -.106749+00                                                                           .385792-02                                                                            -87.930225                               HOYA  C10   -.274406+02                                                                           1.502013                                                                            -.979219-01                                                                           .356850-02                                                                            -87.912933                               HOYA  LAC10 -.272641+02                                                                           1.721032                                                                            -.109430+00                                                                           .401369-02                                                            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                                                                -.113380+00                                                                           .127917-01                                                                            -83.563025                               SCHOTT                                                                               BAFN6 -.875482+01                                                                           1.589874                                                                            -.113357+00                                                                           .129479-01                                                                            -83.483784                               OHARA KZFS4 -.874989+01                                                                           1.614407                                                                            -.125294+00                                                         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                                                              -.125057+00                                                                           .148786-01                                                                            -83.215154                               HOYA  FEL2  -.840377+01                                                                           1.541543                                                                            -.116334+00                                                                           .138431-01                                                                            -83.214029                               HOYA  BAF3  -.836181+01                                                                           1.583573                                                                            -.118003+00                                                                           .141122-01                                                     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                                           LLF2  -.824244+01                                                                           1.541544                                                                            -.116392+00                                                                           .141211-01                                                                            -83.082477                               OHARA LASFO1                                                                               -.819622+01                                                                           1.787175                                                                            -.124162+00                                                                           .151487-01                                                                            -83.043862                               OHARA LLF2  - .819503+01                                                                       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       655401                                                                               -.661610+01                                                                           1.656673                                                                            -.136700+00                                                                           .206617-01                                                                            -81.405012                               OHARA LLF4  -.661334+01                                                                           1.562271                                                                            -.121299+00                                                                           .183416-01                                                                            -81.401454                               OHARA LASFO8                                                                               -.657825+01                             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                                                    -.129208+00                                                                           .245415-01                                                                            -79.245464                               SCHOTT                                                                               TIF2  -.524856+01                                                                           1.533389                                                                            -.119097+00                                                                           .226913-01                                                                            -79.212844                               HOYA  FL1   -.522402+01                                                                           1.574054                                                                            -.128324+00                                                                     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                                                           BASF57                                                                               -.515333+01                                                                           1.652589                                                                            -.130517+00                                                                           .253268-01                                                                            -79.018248                               OHARA LAFO8 -.514535+01                                                                           1.721225                                                                            -.129925+00                                                                           .252510-01                                                                            -79.001640                               CORN-F                                                                         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                                      -70.044926                               CORN-F                                                                               699301                                                                               -.274146+01                                                                           1.700651                                                                            -.180120+00                                                                           .657022-01                                                                            -69.959596                               SCHOTT                                                                               SF18  -.272083+01                                                                           1.723262                                                                            -.185368+00                                                                           .681295-01                                                                            -69.819778                               OHARA SFO1  -.271515+01                                                                           1.721963                                                                            -.184925+00                                                                           .681086-01                                                                            -69.781050                               HOYA  FD1   -.271396+01                                                                           1.719086                                                                            -.183727+00                                                                           .676972-01                                                                            -69.772860                               CHANCE                                                                               717295                                                                               -.271283+01                                                                           1.719090                                                                            -.183675+00                                                                           .677062-01                                                                            -69.765126                               HOYA  FD18  -.271048+01                                                                           1.723258                                                                            -.185364+00                                                                           .683877-01                                                                            -69.749100                               OHARA SF18  -.270769+01                                                                           1.723258                                                                            -.185348+00                                                                           .684524-01                                                                            -69.729891                               OHARA SF15  -.268086+01                                                                           1.700595                                                                            -.179933+00                                                                           .671176-01                                                                            -69.543760                               SCHOTT                                                                               SF15  -.268069+01                                                                           1.700601                                                                            -.180303+00                                                                           .672598-01                                                                            -69.542593                               HOYA  FD15  -.266205+01                                                                           1.700596                                                                            -.180349+00                                                                           .677481-01                                                                            -69.411311                               CHANCE                                                                               699301                                                                               -.265997+01                                                                           1.700596                                                                            -.180374+00                                                                           .678107-01                                                                            -69.396501                               SCHOTT                                                                               SFN64 -.264749+01                                                                           1.707504                                                                            -.179020+00                                                                           .676190-01                                                                            -69.307568                               HOYA  FD20  -.264393+01                                                                           1.721969                                                                            -.185068+00                                                                           .699972-01                                                                            -69.282167                               CORN-F                                                                           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                              SF3   -.257858+01                                                                           1.741863                                                                            -.192066+00                                                                           .744853-01                                                                            -68.803162                               HOYA  FDS2  -.257473+01                                                                           1.730102                                                                            -.188881+00                                                                           .733595-01                                                                            -68.774325                               CORN-F                                                                               836302                                                                               -.257217+01                                                                           1.838371                                                                            -.179459+00                                                                           .697694-01                                                                            -68.755099                               SCHOTT                                                                               LAF9  -.256742+01                                                                           1.797027                                                                            -.190736+00                                                                           .742911-01                                                                            -68.719235                               OHARA SF3   -.256672+01                                                                           1.741853                                                       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                     1.757145                                                                            -.196256+00                                                                           .786372-01                                                                            -68.164672                               SCHOTT                                                                               SF4   -.249448+01                                                                           1.757142                                                                            -.196269+00                                                                           .786815-01                                                                            -68.154840                               OHARA SFO4  -.249191+01                                                                           1.752762                                                                            -.195466+00                                                                           .784402-01                                                                            -68.134485                               CORN-F                                                                               740281                                                                               -.248406+01                                                                           1.741869                                                                            -.192609+00                                                                           .775380-01                                                                            -68.071918                               HOYA  FD4   -.247971+01                                                                           1.757144                                                                            -.196511+00                                                                           .792476-01                                                                            -68.037105                               OHARA SF10  -.246937+01                                                                           1.730063                                                                            -.190214+00                                                                           .770293-01                                                                            -67.953959                               SCHOTT                                                                               SF63  -.246009+01                                                                           1.750313                                                                            -.195287+00                                                                           .793819-01                                                                            -67.878828                               SCHOTT                                                                               SF10  -.245780+01                                                                           1.730067                                                                            -.190535+00                                                                           .775228-01                                                                            -67.860112                               OHARA SF4   -.245714+01                                                                           1.757145                                                                            -.196693+00                                                                           .800493-01                                                                            -67.854864                               CHANCE                                                                               728284                                                                               -.245550+01                                                                           1.730064                                                                            -.190621+00                                                                           .776304-01                                                                            -67.841390                               HOYA  FD10  -.244881+01                                                                           1.730075                                                                            -.191179+00                                                                           .780702-01                                                                            -67.786772                               SCHOTT                                                                               SF61  -.243636+01                                                                           1.752775                                                                            -.196666+00                                                                           .807210-01                                                                            -67.684441                               CORN-F                                                                               755276                                                                               -.241616+01                                                                           1.757143                                                                            -.196095+00                                                                           .811598-01                                                                            - 67.516314                              CORN-F                                                                               762269                                                            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                                                                .840071-01                                                                            -67.162971                               HOYA  FD13  -.237294+01                                                                           1.742663                                                                            -.195067+00                                                                           .822049-01                                                                            -67.148528                               SCHOTT                                                                               SF55  -.236292+01                                                                           1.763801                                                                            -.200664+00                                                                           .849223-01                                                                            -67.061605                               HOYA  FDS5  -.234664+01                                                                           1.763809                                                                            -.200827+00                                                                           .855807-01                                                                            -66.919139                               SCHOTT                                                                               SF13  -.230283+01                                                                           1.742671                                                                            -.195912+00                                                                           .850744-01                                                                            -66.527199                               OHARA SFS3  -.229004+01                                                                           1.786820                                                                            -.206096+00                                                                           .899966-01                                                                            -66.410396                               CHANCE                                                                               805254                                                                               -.227076+01                                                                           1.807417                                                                            -.212405+00                                                                           .935391-01                                                                            -66.232224                               SCHOTT                                                                               TISF1 -.226756+01                                                                           1.675043                                                                            -.187210+00                                                                           .825602-01                                                                            -66.202369                               SCHOTT                                                                               SF56  -.226331+01                                                                           1.786829                                                                            -.207228+00                                                                           .915596-01                                                                            -66.162707                               HOYA  FDS3  -.225619+01                                                                           1.786830                                                                         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                             -.212348+00                                                                           .948645-01                                                                            -65.927739                               OHARA SF6   -.222790+01                                                                           1.807416                                                                            -.212284+00                                                                           .952842-01                                                                            -65.826954                               HOYA  FD6   -.222728+ 01                                                                          1.807421                                                                            -.212038+00                                                                           .952005-01                                                                            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                                      -.205349+00                                                                           .959976-01                                                                            -64.944611                               CHANCE                                                                               785258                                                                               -.213763+01                                                                           1.786872                                                                            -.209608+00                                                                           .980564-01                                                                            -64.929408                               OHARA SFL14 -.212526+01                                                                           1.763851                                                                            -.203665+00    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             HOYA  FD11  -.207581+01                                                                           1.786875                                                                            -.209578+00                                                                           .100962+00                                                                            -64.278046                               OHARA SFL11 -.205802+01                                                                           1.786881                                                                            -.210162+00                                                                           .102119+00                                                                            -64.084599                               HOYA  FDS9  -.205150+01                                                                           1.849166                                                                            -.226160+00                                                                           .110242+00                                                                            -64.013018                               SCHOTT                                                                               SF57  -.205007+01                                                                           1.849171                                                                            -.226170+00                                                                           .110323+00                                                                            -63.997438                               OHARA SFO3  -.204471+01                                                                           1.849159                                                                            -.225686+00                                                                           .110376+00                                                                            -63.938182                               SCHOTT                                                                               TIF6  -.188663+01                                                                           1.617905                                                                            -.174161+00                                                                           .923130-01                                                                            -62.074440                               SCHOTT                                                                               SF58  -.179407+01                                                                           1.920613                                                                            -.249868+00                                                                           .139274+00                                                                            -60.865129                               OHARA SFSO1 -.176690+01                                                                           1.925912                                                                            -.252358+00                                                                           .142826+00                                                                            -60.491622                               OHARA SFS1  -.168930+01                                                                           1.925965                                                                            -.256990+00                                                                           .152128+00                                                                            -59.376132                               SCHOTT                                                                               SF59  -.166253+01                                                                           1.955781                                                                            -.263360+00                                                                           .158409+00                                                                            -58.973417                               SCHOTT                                                                               LF8   -.607584+00                                                                           1.565361                                                                            -.125182+00                                                                           -.206033-01                                                                           -99.346334                               SCHOTT                                                                               FK5   .870540+01                                                                            1.487994                                                                   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                                                        1.487996                                                                            -.792314-01                                                                           -.844991-02                                                                           -96.087499                               SCHOTT                                                                               BK10  .948819+01                                                                            1.498363                                                                            -.830447-01                                                                           -.875243-02                                                                           -96.016432                               HOYA  FC5   .953238+01                                                                            1.487993                                                                    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                                        -.800685-02                                                                           -95.352846                               OHARA BK3   .107940+02                                                                            1.498866                                                                            -.854179-01                                                                           -.791344-02                                                                           -95.292991                               HOYA  FC1   .108373+02                                                                            1.471197                                                                            -.825422-01                                                                           -.761653-02                                                                           -95.272003                               SCHOTT                                                                               PK50  .108546+02                                                                            1.521083                                                                            -.796732-01                                                                           -.734003-02                                                                           -95.263615                               OHARA FK3   .111977+02                                                                            1.465011                                                                            -.842295-01                                                                           -.752205-02                                                                           -95.103216                               OHARA PKS1  .113430+02                                                                            1.517818                                                                            -.797889-01                                                                           -.703421-02                                                                           -95.038185                               OHARA FK1   .115831+02                                                                            1.471198                                                                            -.824182-01                                                                           -.711539-02                                                                           -94.934268                               SCHOTT                                                                               FK3   .117421+02                                                                            1.465014                                                                            -.844297-01                                                                           -.719035-02                                                                           -94.867778                               SCHOTT                                                                               FK54  .120508+02                                                                            1.437351                                                                            -.611697-01                                                                           -.507598-02                                                                           -94.743646                               HOYA  PC2   .121075+02                                                                            1.518788                                                                            -.851916-01                                                                           -.703627-02                                                                           -94.721540                               SCHOTT                                                                               PK2   .122813+02                                                                            1.518790                                                                            -.853260-01                                                                           -.694761-02                                                                           -94.654999                               CORN-F                                                                               465657                                                                               .123644+02                                                                            1.465014                                                                            -.843988-01                                                                           -.682594-02                                                                           -94.623858                               CORN-F                                                                               518651                                                                               .124691+02                                                                            1.518778                                                                            -.852578-01                                                                           -.683755-02                                                                           -94.585223                               HOYA  BSC7  .126988+02                                                                            1.517384                                                                            -.864635-01                                                                           -.680879-02                                                                           -94.502611                               SCHOTT                                                                               PK3   .127884+02                                                                            1.526010                                                                            -.858252-01                                                                           -.671118-02                                                                           -94.471197                               OHARA BK12  .129958+02                                                                            1.519324                                                                            -.860235-01                                                                           -.661931-02                                                                           -94.400107                               HOYA  PC3   .134431+02                                                                            1.526011                                                                            -.858534-01           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                                                                 BK8   .150883+02                                                                            1.520745                                                                            -.870925-01                                                                           -.577220-02                                                                           -93.791827                               HOYA  PCD2  .152105+02                                                                            1.569385                                                                            -.878957-01                                                                           -.577861-02                                                                           -93.761437                               HOYA  FCD10 .152753+02                                                                            1.456866                                                                            -.610293-01                                                                           -.399528-02                                                                           -93.745521                               HOYA  FC3   .153931+02                                                                            1.465015                                                                            -.842094-01                                                                           -.547060-02                                                                           -93.716953                               SCHOTT                                                                               PSK50 .155337+02                                                                            1.558131                                                                            -.823975-01                                                                           -.530444-02     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                        1.552954                                                                            -.873711-01                                                                           -.535026-02                                                                           -93.504190                               SCHOTT                                                                               ZKN7  .164635+02                                                                            1.509073                                                                            -.906253-01                                                                           -.550463-02                                                                           -93.475906                               OHARA ZK7   .165143+02                                                                            1.509077                                                                            -.911171-01                                                                           -.551746-02                                                                           -93.465229                               CORN-F                                                                               510635                                                                               .165281+02                                                                            1.510575                                                                            -.872466-01                                                                           -.527870-02                                                                           -93.462358                               HOYA  BSC1  .167237+02                                                                            1.510675                                                                            -.873966-01                                                                  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    CHANCE                                                                               508612                                                                               .172982+02                                                                            1.508191                                                                            -.906602-01                                                                           -.524101-02                                                                           -93.308551                               SCHOTT                                                                               FK51  .175476+02                                                                            1.486979                                                                            -.655603-01                                                                           -.373614-02                                                                           -93.261635                               SCHOTT                                                                               BK1   .178859+02                                                                            1.510673                                                                            -.873263-01                                                                           -.488241-02                                                                           -93.200073                               CORN-F                                                                               486817                                                                               .192375+02                                                                            1.486475                                                                            -.677108-01                                                                           -.351973-02                                                                           -92.975659                               OHARA BK1   .192502+02                                                                            1.510673                                                                            -.870835-01                                                                           -.452377-02                                                                           -92.973700                               HOYA  PCD3  .194229+02                                                                            1.552952                                                                            -.873730-01                                                                           -.449844-02                                                                           -92.947297                               CHANCE                                                                               569631                                                                               .195119+02                                                                            1.569383                                                                            -.878305-01                                                                           -.450139-02                                                                           -92.933891                               OHARA BK6   .195970+02                                                                            1.531743                                                                            -.886865-01                                                                           -.452553-02                                                                           -92.921178                               OHARA FKO1  .197991+02                                                                            1.497441                                                                            -.678377-01                                                                           -.342631-02                                                                           -92.891407                               HOYA  LACL6 .200987+02                                                                            1.640776                                                                            -.920358-01                                                                           -.457919-02                                                                           -92.848371                               SCHOTT                                                                               BK6   .202252+02                                                                            1.531750                                                                            -.891374-01                                                                           -.440724-02                                                                           -92.830582                               SCHOTT                                                                               FK52  .214118+02                                                                            1.486484                                                                            -.676504-01                                                                           -.315949-02                                                                           -92.673953                               HOYA  BSC6  .214496+02                                                                            1.531749                                                                            -.891886- 01                                                                          -.415805-02                                                                           -92.669247                               OHARA BK2   .216654+02                                                                            1.507162                                                                            -.893758-01                                                                           -.412529-02                                                                           -92.642707                               SCHOTT                                                                               PSK52 .233739+02                                                                            1.603769                                                                            -.846166-01                                                                           -.362012-02                                                                           -92.449769                               OHARA K1    .238599+02                                                                            1.510363                                                                            -.891660-01                                                                           -.373707-02                                                                           -92.399940                               SCHOTT                                                                               K51   .243787+02                                                                            1.505796                                                                            -.930316-01                                                                           -.381609-02                                                                           -92.348915                               CORN-F                                                                               564609                                                                               .245040+02                                                                            1.564676                                                                            -.908447-01                                                                           -.370733-02                                                                           -92.336918                               OHARA SK20  .246088+02                                                                            1.560289                                                                            -.904751-01                                                                           -.367653-02                                                                           -92.326981                               SCHOTT                                                                               SK5   .257685+02                                                                            1.589828                                                                            -.903533-01                                                                           -.350635-02                                                                           -92.222368                               HOYA  C7    .262127+02                                                                            1.511735                                                                            -.914766-01                                                                           -.348978-02                                                                           -92.184742                               CORN-F                                                                               542734                                                                               .262662+02                                                                            1.542737                                                                            -.753406-01                                                                           -.286835-02                                                                           -92.180299                               CORN-F                                                                               518603                                                                               .270791+02                                                                            1.518824                                                                            -.918428-01                                                                           -.339165-02                                                                           -92.114906                               HOYA  BACD5 .275634+02                                                                            1.589827                                                                            -.903801-01                                                                           -.327899-02                                                                           -92.077779                               CHANCE                                                                               589613                                                                               .299354+02                                                                            1.589826                                                                            -.903610-01                                                                           -.301853-02                                                                           -91.913267                               OHARA SK11  .299784+02                                                                            1.564512                                                                            -.911508-01                                                                           -.304055-02                                                                           -91.910527                               HOYA  BACD165                                                                              .301388+02                                                                            1.621164                                                                            -.924448-01                                                                           -.306730-02                                                                           -91.900365                               SCHOTT                                                                               SK20  .311164+02                                                                            1.560294                                                                            -.903862-01                                                                           -.290478-02                                                                           - 91.840705                              HOYA  LACL7 .312404+02                                                                            1.670844                                                                            -.964946-01                                                                           -.308878-02                                                                           -91.833405                               CORN-F                                                                               589612                                                                               .319377+02                                                                            1.589696                                                                            -.904002-01                                                                           -.283052-02                                                                           -91.793402                               CORN-F                                                                               511604                                                                               .322814+02                                                                            1.511613                                                                            -.916603-01                                                                           -.283942-02                                                                           -91.774321                               SCHOTT                                                                               LAK21 .323056+02                                                                            1.641269                                                                            -.920718-01                                                                           -.285003-02                                                                           -91.772991                               HOYA  LAC7  .325405+02                                                                            1.652408                                                                            -.946047-01                                                                           -.290729-02                                                                           -91.760199                               SCHOTT                                                                               PK51  .326136+02                                                                            1.529052                                                                            -.718368-01                                                                           -.220267-02                                                                           -91.756261                               SCHOTT                                                                               K11   .327016+02                                                                            1.500720                                                                            -.900976-01                                                                           -.275514-02                                                                           -91.751530                               HOYA  BACD16                                                                               .333124+02                                                                            1.621155                                                                            -.917236-01                                                                           -.275344-02                                                                           -91.719435                               SCHOTT                                                                               UK50  .336648+02                                                                            1.523199                                                                            -.916329-01                                                                           -.272192-02                                                                           -91.701448                               OHARA LAK7  .338735+02                                                                            1.652403                                                                            -.942570-01                                                                           -.278262-02                                                                           -91.690973                               CHANCE                                                                               697562                                                                               .339006+02                                                                            1.698236                                                                            -.984705-01                                                                           -.290468-02                                                                           -91.689619                               HOYA  BACD11                                                                               .344348+02                                                                            1.564512                                                                            -.909246-01                                                                           -.264048-02                                                                           -91.663421                               SCHOTT                                                                               SK16  .347019+02                                                                            1.621157                                                                            -.917065-01                                                                           -.264269-02                                                                           -91.650625                               OHARA SK14  .349146+02                                                                            1.603833                                                                            -.911086-01                                                                           -.260947-02                                                                           -91.640578                               SCHOTT                                                                               LAK31 .354874+02                                                                            1.697628                                                                            -.980691-01                                                                           -.276349-02                                                                           -91.614111                               OHARA SK5   .360112+02                                                                            1.589827                                                                            -.906992-01                                                                           -.251864-02                                                                           -91.590646                               OHARA LAKO14                                                                               .361383+02                                                                            1.697691                                                                            -.979264-01                                                                           -.270977-02                                                                           -91.585055                               CHANCE                                                                               603606                                                                               .362407+02                                                                            1.603830                                                                            -.912762-01                                                                           -.251861-02                                                                           -91.580577                               SCHOTT                                                                               SK11  .362732+02                                                                            1.564512                                                                            -.909578-01                                                                           -.250757-02                                                                           -91.579158                               SCHOTT                                                                               SK14  .363318+02                                                                            1.603832                                                                            -.912752-01                                                                           -.251227-02                                                                           -91.576617                               CORN-F                                                                               540597                                                                               .366064+02                                                                            1.540160                                                                            -.926842-01                                                                           -.253191-02                                                                           -91.564794                               SCHOTT                                                                               K50   .368838+02                                                                            1.523201                                                                            -.919227-01                                                                           -.249222-02                                                                           -91.553029                               HOYA  BACL3 .369534+02                                                                            1.518973                                                                            -.916143-01                                                                           -.247919-02                                                                           -91.550112                               OHARA LAK6  .375794+02                                                                            1.643293                                                                            -.947540-01                                                                           -.252143-02                                                                           -91.524297                               HOYA  BACD14                                                                               .375934+02                                                                            1.603831                                                                            -.910845-01                                                                           -.242289-02                                                                           -91.523735                               CHANCE                                                                               564608                                                                               .389388+02                                                                            1.564511                                                                            -.909646-01                                                                           -.233609-02                                                                           -91.471107                               HOYA  LAC6  .390387+02                                                                            1.643302                                                                            -.953703-01                                                                           -.244297-02                                                                           -91.467346                               CHANCE                                                                               620603                                                                               .397432+02                                                                            1.621154                                                                            -.916871-01                                                                           -.230699-02                                                                           -91.441346                               CORN-F                                                                               620603                                                                               .404871+02                                                                            1.621151                                                                            -.916950-01                                                                           -.226480-02                                                                           -91.414876                               SCHOTT                                                                               KZFSN2                                                                               .414694+02                                                                            1.559109                                                                            -.102222+00                                                                           -.246500-02                                                                           -91.381373                               HOYA  LAC11 .416691+02                                                                            1.659130                                                                            -.965613-01                                                                           - .231733-02                                                                          -91.374750                               CHANCE                                                                               519604                                                                               .422941+02                                                                            1.519611                                                                            -.915426-01                                                                           -.216443-02                                                                           -91.354446                               CORN-F                                                                               641601                                                                               .439338+02                                                                            1.641267                                                                            -.919855-01                                                                           -.209373-02                                                                           -91.303914                               HOYA  LAC15 .444092+02                                                                            1.697695                                                                            -.985518-01                                                                           -.221917-02                                                                           -91.289957                               SCHOTT                                                                               K7    .449551+02                                                                            1.511734                                                                            -.915512-01                                                                           -.203650-02                                                                           -91.274299                               CORN-F                                                                               652585                                                                               .450442+02                                                                            1.652407                                                                            -.944992-01                                                                           -.209792-02                                                                           -91.271780                               SCHOTT                                                                               BALKN3                                                                               .450955+02                                                                            1.519112                                                                            -.917996-01                                                                           -.203567-02                                                                           -91.270334                               CORN-F                                                                               697562                                                                               .451971+02                                                                            1.697899                                                                            -.983871-01                                                                           -.217685-02                                                                           -91.267482                               OHARA BALK1 .456965+02                                                                            1.527056                                                                            -.921451-01                                                                           -.201646-02                                                                           -91.253633                               HOYA  LAC14 .466141+02                                                                            1.697713                                                                            -.997339-01                                                                           -.213956-02                                                                           -91.228959                               CHANCE                                                                               697554                                                                               .470812+02                                                                            1.697710                                                                            -.998106-01                                                                           -.211997-02                                                                           -91.216775                               HOYA  BACD12                                                                               .471840+02                                                                            1.583840                                                                            -.928865-01                                                                           -.196860-02                                                                           -91.214122                               SCHOTT                                                                               PSK53 .482711+02                                                                            1.620848                                                                            -.870163-01                                                                           -.180266-02                                                                           -91.186790                               OHARA LAK14 .486738+02                                                                            1.697706                                                                            -.994448-01                                                                           -.204309-02                                                                           -91.176973                               OHARA BALK3 .497837+02                                                                            1.518974                                                                            -.917072-01                                                                           -.184211-02                                                                           -91.150737                               HOYA  LAC9  .522504+02                                                                            1.691912                                                                            -.101030+00                                                                           -.193358-02                                                                           -91.096432                               HOYA  TAC1  .543995+02                                                                            1.726985                                                                            -.103520+00                                                                           -.190295-02                                                                           -91.053118                               OHARA SK16  .559137+02                                                                            1.621156                                                                            -.917056-01                                                                           -.164013-02                                                                           -91.024611                               SCHOTT                                                                               LAKN14                                                                               .576174+02                                                                            1.697711                                                                            -.997837-01                                                                           -.173183-02                                                                           -90.994316                               OHARA PSKO1 .586782+02                                                                            1.617713                                                                            -.880364-01                                                                           -.150033-02                                                                           -90.976349                               OHARA SK12  .596787+02                                                                            1.583841                                                                            -.930734-01                                                                           -.155957-02                                                                           -90.959978                               OHARA BAK2  .633248+02                                                                            1.540616                                                                            -.927256-01                                                                           -.146429-02                                                                           -90.904719                               CORN-F                                                                               523594                                                                               .679816+02                                                                            1.523633                                                                            -.929572-01                                                                           -.136739-02                                                                           -90.842753                               OHARA K7    .690021+02                                                                            1.511733                                                                            -.914017-01                                                                           -.132462-02                                                                           -90.830289                               OHARA K50   .721590+02                                                                            1.523196                                                                            -.921678-01                                                                           -.127729-02                                                                           -90.793972                               HOYA  BACD7 .747049+02                                                                            1.608032                                                                            -.928925-01                                                                           -.124346-02                                                                           -90.766915                               CHANCE                                                                               524592                                                                               .747616+02                                                                            1.524639                                                                            -.933783-01                                                                           -.124901-02                                                                           -90.766332                               SCHOTT                                                                               LAKN7 .749037+02                                                                            1.652407                                                                            -.944376-01                                                                           -.126079-02                                                                           -90.764882                               CHANCE                                                                               652585                                                                               .757398+02                                                                            1.652404                                                                            -.944566-01                                                                           -.124712-02                                                                           -90.756438                               CORN-F                                                                               697554                                                                               .760379+02                                                                            1.697915                                                                            -.998003-01                                                                           -.131251-02                                                                           -90.753474                               HOYA  LACL1 .780667+02                                                                            1.641669                                                                            -.972321-01                                                                           -.124550-02                                                                           -90.733893                               SCHOTT                                                                               SK7   .798809+02                                                                            1.608030                                                                            - .929450-01                                                                          -.116354-02                                                                           -90.717224                               SCHOTT                                                                               SK12  .836981+02                                                                            1.583842                                                                            -.929417-01                                                                           -.111044-02                                                                           -90.684521                               SCHOTT                                                                               BALK1 .871857+02                                                                            1.527057                                                                            -.920710-01                                                                           -.105603-02                                                                           -90.657138                               SCHOTT                                                                               BAK50 .875947+02                                                                            1.568446                                                                            -.952900-01                                                                           -.108785-02                                                                           -90.654072                               HOYA  BAC5  .890026+02                                                                            1.557397                                                                            -.942542-01                                                                           -.105900-02                                                                           -90.643723                               CORN-F                                                                               613585                                                                               .897103+02                                                                            1.613451                                                                            -.944049-01                                                                           -.105233-02                                                                           -90.638648                               CHANCE                                                                               607595                                                                               .897382+02                                                                            1.608030                                                                            -.929421-01                                                                           -.103570-02                                                                           -90.638449                               SCHOTT                                                                               TIK1  .924535+02                                                                            1.479281                                                                            -.941893-01                                                                           -.101877-02                                                                           -90.619698                               SCHOTT                                                                               K5    .939519+02                                                                            1.523126                                                                            -.929097-01                                                                           -.988908-03                                                                           -90.609818                               HOYA  ZNC1  .957631+02                                                                            1.533814                                                                            -.951004-01                                                                           -.993080-03                                                                           -90.598286                               OHARA LAK18 .997389+02                                                                            1.730123                                                                            -.101009+00                                                                           -.101274-02                                                                           -90.574442                               SCHOTT                                                                               BAK2  .103271+03                                                                            1.540615                                                                            -.925331-01                                                                           -.896020-03                                                                           -90.554791                               CORN-F                                                                               604640                                                                               .109857+03                                                                            1.604192                                                                            -.863228-01                                                                           -.785775-03                                                                           -90.521534                               OHARA SK7   .110102+03                                                                            1.608032                                                                            -.930232-01                                                                           -.844881-03                                                                           -90.520373                               SCHOTT                                                                               LAK11 .110909+03                                                                            1.659134                                                                            -.965050-01                                                                           -.870126-03                                                                           -90.516586                               OHARA LAKO2 .114422+03                                                                            1.670846                                                                            -.963556-01                                                                           -.842108-03                                                                           -90.500729                               OHARA SK3   .116368+03                                                                            1.609560                                                                            -.937132-01                                                                           -.805320-03                                                                           -90.492356                               OHARA PSKO2 .125723+03                                                                            1.618706                                                                            -.871342-01                                                                           -.693063-03                                                                           -90.455719                               HOYA  LAC8  .128627+03                                                                            1.713957                                                                            -.102430+00                                                                           -.796330-03                                                                           -90.445430                               HOYA  C5    .152053+03                                                                            1.523124                                                                            -.929279-01                                                                           -.611157-03                                                                           -90.376810                               HOYA  BACD15                                                                               .152779+03                                                                            1.623771                                                                            -.950835-01                                                                           -.622358-03                                                                           -90.375017                               CHANCE                                                                               669574                                                                               .156946+03                                                                            1.669663                                                                            -.962691-01                                                                           -.613390-03                                                                           -90.365062                               SCHOTT                                                                               SK3   .158337+03                                                                            1.609558                                                                            -.937522-01                                                                           -.592104-03                                                                           -90.361854                               HOYA  BACL1 .165873+03                                                                            1.527052                                                                            -.919471-01                                                                           -.554323-03                                                                           -90.345415                               SCHOTT                                                                               K3    .177587+03                                                                            1.518886                                                                            -.936677-01                                                                           -.527447-03                                                                           -90.322632                               CHANCE                                                                               643580                                                                               .179454+03                                                                            1.643299                                                                            -.953194-01                                                                           -.531164-03                                                                           -90.319275                               OHARA K5    .192279+03                                                                            1.523127                                                                            -.924409-01                                                                           -.480765-03                                                                           -90.297979                               HOYA  BAC2  .198263+03                                                                            1.540616                                                                            -.925125-01                                                                           -.466615-03                                                                           -90.288986                               CORN-F                                                                               609590                                                                               .208907+03                                                                            1.609502                                                                            -.935670-01                                                                           -.447888-03                                                                           -90.274261                               CHANCE                                                                               713538                                                                               .211681+03                                                                            1.713958                                                                            -.102639+00                                                                           -.484875-03                                                                           -90.270667                               OHARA LAK9  .222992+03                                                                            1.691914                                                                            -.100754+00                                                                           -.451830-03                                                                           -90.256941                               CORN-F                                                                               623581                                                                               .225515+ 03                                                                           1.623780                                                                            -.951191-01                                                                           -.421786-03                                                                           -90.254064                               HOYA  ADF1  .228995+03                                                                            1.565769                                                                            -.104320+00                                                                           -.455555-03                                                                           -90.250203                               SCHOTT                                                                               LAK23 .235819+03                                                                            1.669660                                                                            -.962547-01                                                                           -.408172-03                                                                           -90.242963                               SCHOTT                                                                               LAKN6 .237437+03                                                                            1.643302                                                                            -.952892-01                                                                           -.401324-03                                                                           -90.241308                               SCHOTT                                                                               LAK9  .257817+03                                                                            1.691917                                                                            -.100967+00                                                                           -.391624-03                                                                           -90.222234                               SCHOTT                                                                               LAK8  .280438+03                                                                            1.713962                                                                            -.102629+00                                                                           -.365959-03                                                                           -90.204307                               SCHOTT                                                                               SK4   .306677+03                                                                            1.613477                                                                            -.942118-01                                                                           -.307202-03                                                                           -90.186827                               HOYA  BACD4 .371294+03                                                                            1.613475                                                                            -.942747-01                                                                           -.253908-03                                                                           -90.154312                               HOYA  BACD3 .416634+03                                                                            1.609561                                                                            -.938263-01                                                                           -.225201-03                                                                           -90.137521                               CHANCE                                                                               613586                                                                               .428948+03                                                                            1.613475                                                                            -.942101-01                                                                           -.219630-03                                                                           -90.133572                               CHANCE                                                                               658573                                                                               .484001+03                                                                            1.659131                                                                            -.964698-01                                                                           -.199317-03                                                                           -90.118379                               OHARA K3    .569286+03                                                                            1.518865                                                                            -.937249-01                                                                           -.164636-03                                                                           -90.100645                               SCHOTT                                                                               K4    .572325+03                                                                            1.519606                                                                            -.962542-01                                                                           -.168181-03                                                                           -90.100111                               CORN-F                                                                               678555                                                                               .581978+03                                                                            1.678889                                                                            -.994888-01                                                                           -.170950-03                                                                           -90.098450                               CHANCE                                                                               623581                                                                               .670154+03                                                                            1.623764                                                                            -.951456-01                                                                           -.141976-03                                                                           -90.085497                               HOYA  TAC2  .914435+03                                                                            1.742017                                                                            -.104919+00                                                                           -.114737-03                                                                           -90.062657                               CORN-F                                                                               573575                                                                               .116669+05                                                                            1.573219                                                                            -.961053-01                                                                           -.823742-05                                                                           -90.004910                               __________________________________________________________________________

                                      TABLE II                                     __________________________________________________________________________     LEAST SQUARES FITTED COEFFICIENTS                                                                 ORDER                                                                               BASE WVL                                                                              MIN WVL                                                                              MAX WVL                                                      3.0  0.5890 0.360 1.014                                     CAT   NAME  ETA1/ETA2                                                                              INDEX  ETA1   ETA2    ETA3   ETA2/ETA3                                                                             ETA1/ETA3              __________________________________________________________________________     CORN-F                                                                               558525                                                                               -1050.530                                                                              1.557434                                                                              -.974752-01                                                                           .927867-04                                                                             -.390622-02                                                                           -.024  24.954                 SCHOTT                                                                               LASF11                                                                               -752.430                                                                               1.801525                                                                              -.115759+00                                                                           .153847-03                                                                             .524469-02                                                                            .029   -22.072                OHARA BAF5  -594.102                                                                               1.607198                                                                              -.104136+00                                                                           .175283-03                                                                             .554173-02                                                                            .032   -18.791                SCHOTT                                                                               BAFN11                                                                               -377.046                                                                               1.666620                                                                              -.105738+00                                                                           .280437-03                                                                             .413480-02                                                                            .068   -25.573                HOYA  LAFL4 -351.383                                                                               1.712584                                                                              -.118572+00                                                                           .337443-03                                                                             .277293-02                                                                            .122   -42.760                HOYA  BAF10 -344.527                                                                               1.669924                                                                              -.108569+00                                                                           .315124-03                                                                             .207762-02                                                                            .152   -52.256                OHARA SSK5  -323.740                                                                               1.658342                                                                              -.100614+00                                                                           .310787-03                                                                             .104009-01                                                                            .030   -9.674                 SCHOTT                                                                               BAF5  -263.952                                                                               1.607199                                                                              -.103629+00                                                                           .392606-03                                                                             .601944-02                                                                            .065   -17.216                SCHOTT                                                                               LAF23 -245.436                                                                               1.688900                                                                              -.103472+00                                                                           .421584-03                                                                             .795724-02                                                                            .053   -13.003                HOYA  BAF6  -234.257                                                                               1.588914                                                                              - .105634+00                                                                          .450934-03                                                                             .482195-02                                                                            .094   -21.907                SCHOTT                                                                               LAF20 -231.441                                                                               1.682372                                                                              -.106185+00                                                                           .458798-03                                                                             .612522-02                                                                            .075   -17.336                HOYA  NBFD11                                                                               -231.215                                                                               1.785765                                                                              -.116614+00                                                                           .504353-03                                                                             .718313-02                                                                            .070   -16.234                OHARA BAF2  -203.299                                                                               1.569565                                                                              -.103725+00                                                                           .510213-03                                                                             .582092-02                                                                            .088   -17.819                HOYA  BAF5  -189.571                                                                               1.607202                                                                              -.103909+00                                                                           .548126-03                                                                             .549668-02                                                                            .100   -18.904                CORN-F                                                                               548457                                                                               -171.777                                                                               1.548010                                                                              -.112116+00                                                                           .652683-03                                                                             .238165-02                                                                            .274   -47.075                OHARA LAFO1 -154.399                                                                               1.699891                                                                              -.106472+00                                                                           .689593-03                                                                             .486976-02                                                                            .142   -21.864                OHARA LAFO9 -139.242                                                                               1.696897                                                                              -.105459+00                                                                           .757380-03                                                                             .640845-02                                                                            .118   -16.456                SCHOTT                                                                               LAF10 -118.034                                                                               1.784300                                                                              -.116661+00                                                                           .988369-03                                                                             .515218-02                                                                            .192   -22.643                SCHOTT                                                                               BAF53 -112.012                                                                               1.669925                                                                              -.108658+00                                                                           .970051-03                                                                             .386597-02                                                                            .251   -28.106                CHANCE                                                                               548456                                                                               -107.436                                                                               1.547602                                                                              -.112271+00                                                                           .104501-02                                                                             .241302-02                                                                            .433   -46.527                OHARA BAF6  -100.221                                                                               1.588912                                                                              -.105204+00                                                                           .104972-02                                                                             .566316-02                                                                            .185   -18.577                HOYA  LAFL3 -97.646 1.700195                                                                              -.106981+00                                                                           .109560-02                                                                             .639820-02                                                                            .171   -16.720                HOYA  FEL3  -89.849 1.560039                                                                              -.108708+00                                                                           .120990-02                                                                             .310242- 02                                                                           .390   -35.040                HOYA  BAF21 -89.124 1.664119                                                                              -.104787+00                                                                           .117575-02                                                                             .578126-02                                                                            .203   -18.125                OHARA LAF3  -81.326 1.716894                                                                              -.106686+00                                                                           .131184-02                                                                             .806100-02                                                                            .163   -13.235                OHARA BAF1  -80.823 1.556816                                                                              -.105401+00                                                                           .130410-02                                                                             .509190-02                                                                            .256   -20.700                OHARA BAF11 -77.909 1.666617                                                                              -.105827+00                                                                           .135835-02                                                                             .544010-02                                                                            .250   -19.453                SCHOTT                                                                               LLF7  -69.953 1.548743                                                                              -.112723+00                                                                           .161141-02                                                                             .233046-02                                                                            .691   -48.369                OHARA LLF3  -69.160 1.560042                                                                              -.108884+00                                                                           .157438-02                                                                             .372003-02                                                                            .423   -29.270                HOYA  BAF9  -68.508 1.643179                                                                              -.106698+00                                                                           .155745-02                                                                             .564904-02                                                                            .276   -18.888                SCHOTT                                                                               KZFSN5                                                                               -65.662 1.653997                                                                              -.129255+00                                                                           .196849-02                                                                             -.310441-02                                                                           -.634  41.636                 SCHOTT                                                                               LLF1  -63.655 1.548053                                                                              -.111850+00                                                                           .175712-02                                                                             .253850-02                                                                            .692   -44.061                OHARA LASFO1                                                                               -63.641 1.785766                                                                              -.115810+00                                                                           .181975-02                                                                             .673988-02                                                                            .270   -17.183                SCHOTT                                                                               LAFN3 -60.968 1.716888                                                                              -.106539+00                                                                           .174745-02                                                                             .955062-02                                                                            .183   -11.155                CORN-F                                                                               670471                                                                               -58.527 1.669901                                                                              -.108558+00                                                                           .185484-02                                                                             .814704-02                                                                            .228   -13.325                HOYA  FF2   -57.558 1.532476                                                                              -.111413+00                                                                           .193568-02                                                                             -.109276-01                                                                           -.177  10.196                 SCHOTT                                                                               BAFN10                                                                               -54.216 1.669927                                                                              -.108512+00                                                                           .200146-02                                                                             .413130-02                                                                            .484   -26.266                OHARA LASFO6                                                                               -53.532 1.840854                                                                              -.118293+00                                                                           .220977-02                                                                             .901010-02                                                                            .245   -13.129                OHARA LLF1  -53.174 1.548053                                                                              -.111719+00                                                                           .210102-02                                                                             .330115-02                                                                            .636   -33.843                OHARA LLF7  -52.684 1.548601                                                                              -.112212+00                                                                           .212992-02                                                                             .302928-02                                                                            .703   -37.042                HOYA  FEL1  -51.632 1.548049                                                                              -.111508+00                                                                           .215968-02                                                                             .364425-02                                                                            .593   -30.598                SCHOTT                                                                               TIF2  -51.036 1.532474                                                                              -.111177+00                                                                           .217838-02                                                                             -.947587-02                                                                           -.230  11.733                 HOYA  LAF2  -49.984 1.743876                                                                              -.114058+00                                                                           .228190-02                                                                             .341337-02                                                                            .669   -33.415                HOYA  TAFD5 -48.089 1.834857                                                                              -.119028+00                                                                           .247515-02                                                                             .924471-02                                                                            .268   -12.875                SCHOTT                                                                               BAF9  -45.451 1.643179                                                                              -.106557+00                                                                           .234445-02                                                                             .706099-02                                                                            .332   -15.091                OHARA KZFS5 -44.759 1.653994                                                                              -.128928+00                                                                           .288054-02                                                                             -.161015-02                                                                           -1.789 80.072                 HOYA  FEL7  -44.140 1.548741                                                                              -.112447+00                                                                           .254752-02                                                                             .376192-02                                                                            .677   -29.891                OHARA BAF9  -40.296 1.643181                                                                              -.106837+00                                                                           .265129-02                                                                             .686698-02                                                                            .386   -15.558                OHARA BAF10 -39.937 1.669925                                                                              -.108092+00                                                                           .270654-02                                                                             .493908-02                                                                            .548   -21.885                OHARA LASFO5                                                                               -39.536 1.834664                                                                              -.119657+00                                                                           .302656-02                                                                             .845778-02                                                                            .358   -14.148                SCHOTT                                                                               LLF4  -37.349 1.561290                                                                              -.113002+00                                                                           .302554-02                                                                             .337006-02                                                                            .898   -33.531                HOYA  LAFL1 -36.213 1.685902                                                                              -.103528+00                                                                           .285890-02                                                                             .119049-01                                                                            .240   -8.696                 HOYA  NBFD1 -34.398 1.788869                                                                              -.119434+00                                                                           .347212-02                                                                             .725389-02                                                                            .479   -16.465                HOYA  BAF11 -34.301 1.666619                                                                              -.105769+00                                                                           .308354-02                                                                             .795536-02                                                                            .388   -13.295                HOYA  BAF8  -33.837 1.623642                                                                              -.108554+00                                                                           .320811-02                                                                             .707055-02                                                                            .454   -15.353                HOYA  FEL4  -33.462 1.561288                                                                              -.113039+00                                                                           .337814-02                                                                             .462977-02                                                                            .730   -24.416                OHARA LLF8  -32.139 1.532480                                                                              -.111282+00                                                                           .346255-02                                                                             -.794041-02                                                                           -.436  14.015                 OHARA LLF4  -30.955 1.561288                                                                              -.113115+00                                                                           .365418-02                                                                             .425506-02                                                                            .859   -26.584                SCHOTT                                                                               BAF8  -29.261 1.623643                                                                              -.108607+00                                                                           .371162-02                                                                             .742686-02                                                                            .500   -14.623                OHARA BAF8  -28.362 1.623643                                                                              -.108774+00                                                                           .383526-02                                                                             .718593-02                                                                            .534   -15.137                SCHOTT                                                                               BAF3  -27.953 1.582578                                                                              -.109864+00                                                                           .393029-02                                                                             .656866-02                                                                            .598   -16.725                CORN-F                                                                               667484                                                                               -26.737 1.666597                                                                              -.105346+00                                                                           .394003-02                                                                             .112015-01                                                                            .352   -9.405                 HOYA  BAF3  -24.490 1.582578                                                                              - .109451+00                                                                          .446919-02                                                                             .820470-02                                                                            .545   -13.340                HOYA  BAF7  -24.292 1.607917                                                                              -.110450+00                                                                           .454684-02                                                                             .674832-02                                                                            .674   -16.367                SCHOTT                                                                               BAF52 -23.590 1.608493                                                                              -.109821+00                                                                           .465544-02                                                                             .636994-02                                                                            .731   -17.240                CORN-F                                                                               606439                                                                               -23.512 1.605308                                                                              -.116186+00                                                                           .494161-02                                                                             .131351-02                                                                            3.762  -88.454                HOYA  NBFD12                                                                               -23.111 1.799364                                                                              -.120496+00                                                                           .521386-02                                                                             .102631-01                                                                            .508   -11.741                HOYA  TAFD2 -22.942 1.837551                                                                              -.121732+00                                                                           .530614-02                                                                             .118741-01                                                                            .447   -10.252                OHARA LAF2  -22.723 1.743877                                                                              -.114027+00                                                                           .501816-02                                                                             .900490-02                                                                            .557   -12.663                OHARA BAF3  -22.658 1.582579                                                                              -.110049+00                                                                           .485696-02                                                                             .780848-02                                                                            .622   -14.094                OHARA LASFO2                                                                               -22.164 1.799377                                                                              -.120641+00                                                                           .544307-02                                                                             .803180-02                                                                            .678   -15.020                OHARA BAFO1 -20.498 1.682384                                                                              -.113645+00                                                                           .554431-02                                                                             -.319651-02                                                                           -1.734 35.553                 SCHOTT                                                                               BAF12 -19.769 1.639197                                                                              -.112813+00                                                                           .570667-02                                                                             .570551-02                                                                            1.000  -19.773                SCHOTT                                                                               BASF52                                                                               -19.185 1.701684                                                                              -.124359+00                                                                           .648195-02                                                                             .301104-02                                                                            2.153  -41.301                SCHOTT                                                                               BAF51 -19.184 1.652135                                                                              -.113428+00                                                                           .591270-02                                                                             .806229-02                                                                            .733   -14.069                SCHOTT                                                                               LF8   -19.124 1.564341                                                                              -.116535+00                                                                           .609355-02                                                                             .539743-02                                                                            1.129  -21.591                SCHOTT                                                                               BAF50 -19.106 1.682613                                                                              -.114474+00                                                                           .599166-02                                                                             .852320-02                                                                            .703   -13.431                HOYA  BAF13 -18.841 1.668807                                                                              -.113565+00                                                                           .602752-02                                                                             .568205-02                                                                            1.061  -19.987                HOYA  FL6   -18.794 1.567227                                                                              -.119041+00                                                                           .633390-02                                                                             .410144-02                                                                            1.544  -29.024                HOYA  LAFL5 -18.320 1.719877                                                                              -.115979+00                                                                           .633086-02                                                                             .685298-02                                                                            .924   -16.924                OHARA LF8   -18.227 1.564346                                                                              -.116391+00                                                                           .638568-02                                                                             .576779-02                                                                            1.107  -20.180                SCHOTT                                                                               BAF13 -17.549 1.668811                                                                              -.113273+00                                                                           .645484-02                                                                             .633754-02                                                                            1.019  -17.873                OHARA LAFO11                                                                               -17.479 1.719887                                                                              -.110611+00                                                                           .632831-02                                                                             .120600-01                                                                            .525   -9.172                 SCHOTT                                                                               LAFN2 -17.470 1.743879                                                                              -.113731+00                                                                           .651028-02                                                                             .113526-01                                                                            .573   -10.018                SCHOTT                                                                               LAF25 -17.388 1.784133                                                                              -.123364+00                                                                           .709495-02                                                                             .450414-02                                                                            1.575  -27.389                HOYA  BAF22 -17.185 1.682384                                                                              -.114314+00                                                                           .665199-02                                                                             .665086-02                                                                            1.000  -17.188                OHARA BAF13 -16.928 1.668813                                                                              -.113221+00                                                                           .668836-02                                                                             .660533-02                                                                            1.013  -17.141                OHARA LASFO7                                                                               -16.700 1.862845                                                                              -.122676+00                                                                           .734572-02                                                                             .118988-01                                                                            .617   -10.310                SCHOTT                                                                               LF6   -16.551 1.567223                                                                              -.118934+00                                                                           .718604-02                                                                             .544410-02                                                                            1.320  -21.846                CHANCE                                                                               567428                                                                               -16.328 1.567221                                                                              -.118977+00                                                                           .728671-02                                                                             .571840-02                                                                            1.274  -20.806                OHARA BAFO3 -16.241 1.685664                                                                              -.115724+00                                                                           .712537-02                                                                             .376098-02                                                                            1.895  -30.770                HOYA  LAFL6 -16.040 1.719874                                                                              -.120916+00                                                                           .753820-02                                                                             .637544-02                                                                            1.182  -18.966                SCHOTT                                                                               LAFN8 -15.804 1.735068                                                                              -.122469+00                                                                           .774926-02                                                                             .579672-02                                                                            1.337  -21.127                HOYA  BAF20 -15.734 1.685663                                                                              -.115826+00                                                                           .736131-02                                                                             .575338-02                                                                            1.279  -20.132                HOYA  BAF12 -15.690 1.639198                                                                              -.112862+00                                                                           .719328-02                                                                             .669007-02                                                                            1.075  -16.870                HOYA  NBFD2 -15.464 1.797057                                                                              -.123831+00                                                                           .800752-02                                                                             .109175-01                                                                            .733   -11.342                HOYA  FL4   -15.210 1.578346                                                                              -.121940+00                                                                           .801715-02                                                                             .442920-02                                                                            1.810  -27.531                OHARA BAF12 -14.896 1.639196                                                                              -.113404+00                                                                           .761280-02                                                                             .909745-02                                                                            .837   -12.465                CHANCE                                                                               573426                                                                               -14.766 1.572989                                                                              -.119598+00                                                                           .809957-02                                                                             .675199-02                                                                            1.200  -17.713                SCHOTT                                                                               LAF26 -14.692 1.745837                                                                              -.127243+00                                                                           .866046-02                                                                             .373026-02                                                                            2.322  -34.111                SCHOTT                                                                               LASFN3                                                                               -14.611 1.807865                                                                              -.124911+00                                                                           .854927-02                                                                             .843706-02                                                                            1.013  -14.805                OHARA LASFO3                                                                               -14.547 1.805953                                                                              -.124719+00                                                                           .857340-02                                                                             .849751-02                                                                            1.009  -14.677                CHANCE                                                                               582421                                                                               -14.544 1.582047                                                                              -.121034+00                                                                           .832200-02                                                                             .618517-02                                                                            1.345  -19.568                SCHOTT                                                                               LF1   -14.420 1.572991                                                                              -.119537+00                                                                           .828961-02                                                                             .659340-02                                                                            1.257  -18.130                OHARA LF6   -14.084 1.567226                                                                              -.118862+00                                                                           .843972-02                                                                             .734061-02                                                                            1.150  -16.192                HOYA  BAFD15                                                                               -14.064 1.701858                                                                              -.126284+00                                                                           .897935-02                                                                             .846666-02                                                                            1.061  -14.916                OHARA LF3   -14.044 1.582050                                                                              -.120905+00                                                                           .860920-02                                                                             .582093-02                                                                            1.479  -20.771                OHARA LAFO7 -13.840 1.712728                                                                              -.117774+00                                                                           .850955-02                                                                             .693729-02                                                                            1.227  -16.977                SCHOTT                                                                               LF3   -13.696 1.582050                                                                              -.120955+00                                                                           .883166-02                                                                             .613708-02                                                                            1.439  -19.709                HOYA  NBFD13                                                                               -13.631 1.805957                                                                              -.124868+00                                                                           .916079-02                                                                             .111698-01                                                                            .820   -11.179                OHARA LAFO2 -13.622 1.719879                                                                              -.116357+00                                                                           .854199-02                                                                             .881269-02                                                                            .969   -13.203                SCHOTT                                                                               LASF31                                                                               -13.091 1.880513                                                                              -.123721+00                                                                           .945100-02                                                                             .172923-01                                                                            .547   -7.155                 CHANCE                                                                               606439                                                                               -13.083 1.605513                                                                              -.115836+00                                                                           .885368-02                                                                             .978139-02                                                                            .905   -11.842                SCHOTT                                                                               LF4   -13.066 1.578349                                                                              -.122313+00                                                                           .936105-02                                                                             .616887-02                                                                            1.517  -19.827                SCHOTT                                                                               BAF4  -12.972 1.605519                                                                              -.115760+00                                                                           .892355-02                                                                             .960315-02                                                                            .929   -12.054                OHARA LF1   -12.902 1.572993                                                                              -.119457+00                                                                           .925863-02                                                                             .779907-02                                                                            1.187  -15.317                SCHOTT                                                                               LF7   -12.737 1.574910                                                                              -.122605+00                                                                           .962558-02                                                                             .651416-02                                                                            1.478  -18.821                HOYA  FL3   -12.685 1.582049                                                                              -.121048+00                                                                           .954243-02                                                                             .652345-02                                                                            1.463  -18.556                HOYA  BAF4  -12.677 1.605519                                                                              -.115915+00                                                                           .914392-02                                                                             .992630-02                                                                            .921   -11.678                SCHOTT                                                                               KZFSN7                                                                               -12.518 1.680506                                                                              -.136829+00                                                                           .109307-01                                                                             -.137634-02                                                                           -7.942 99.416                 CORN-F                                                                               696365                                                                               -12.487 1.695850                                                                              -.139493+00                                                                           .111714-01                                                                             .324357-02                                                                            3.444  -43.006                SCHOTT                                                                               TIF3  -12.443 1.547555                                                                              -.120418+00                                                                           .967782-02                                                                             -.726416-02                                                                           -1.332 16.577                 OHARA LASF13                                                                               -12.438 1.804250                                                                              -.128526+00                                                                           .103335-01                                                                             .786269-02                                                                            1.314  -16.346                HOYA  FL1   -12.431 1.572992                                                                              -.119456+00                                                                           .960991-02                                                                             .818230-02                                                                            1.174  -14.599                OHARA LF7   -12.209 1.574905                                                                              -.122511+00                                                                           .100343-01                                                                             .729838-02                                                                            1.375  -16.786                OHARA LASFO8                                                                               -12.208 1.882839                                                                              -.124656+00                                                                           .102113-01                                                                             .165761-01                                                                            .616   -7.520                 HOYA  FL7   -11.928 1.574909                                                                              -.122437+00                                                                           .102643-01                                                                             .792764-02                                                                            1.295  -15.444                CHANCE                                                                               700412                                                                               -11.906 1.699880                                                                              -.123488+00                                                                           .103720-01                                                                             .760763-02                                                                            1.363  -16.232                OHARA BAF4  -11.837 1.605519                                                                              -.116199+00                                                                           .981703-02                                                                             .984560-02                                                                            .997   -11.802                SCHOTT                                                                               LASF15                                                                               -11.816 1.877832                                                                              -.133184+00                                                                           .112711-01                                                                             .102777-01                                                                            1.097  -12.959                SCHOTT                                                                               BASF57                                                                               -11.777 1.651362                                                                              -.121285+00                                                                           .102987-01                                                                             .661943-02                                                                            1.556  -18.322                HOYA  NBFD3 -11.652 1.804355                                                                              -.128259+00                                                                           .110078-01                                                                             .119855-01                                                                            .918   -10.701                HOYA  BAFD5 -11.534 1.603128                                                                              -.119932+00                                                                           .103984-01                                                                             .846102-02                                                                            1.229  -14.175                OHARA LF4   -11.473 1.578349                                                                              -.122479+00                                                                           .106757-01                                                                             .766185-02                                                                            1.393  -15.986                CHANCE                                                                               581409                                                                               -11.410 1.581340                                                                              -.124300+00                                                                           .108938-01                                                                             .764572-02                                                                            1.425  -16.257                SCHOTT                                                                               BASF5 -11.378 1.603127                                                                              -.119614+00                                                                           .105128-01                                                                             .927692-02                                                                            1.133  -12.894                CORN-F                                                                               581408                                                                               -11.181 1.581292                                                                              -.124582+00                                                                           .111423-01                                                                             .857334-02                                                                            1.300  -14.531                HOYA  FL2   -11.000 1.589106                                                                              -.123975+00                                                                           .112707-01                                                                             .791513-02                                                                            1.424  -15.663                OHARA LF5   -10.756 1.581335                                                                              -.124689+00                                                                           .115920-01                                                                             .795933-02                                                                            1.456  -15.666                OHARA LAFO8 -10.587 1.719874                                                                              -.120601+00                                                                           .113914-01                                                                             .844621-02                                                                            1.349  -14.279                OHARA BASF5 -10.543 1.603126                                                                              -.119991+00                                                                           .113807-01                                                                             .105922-01                                                                            1.074  -11.328                SCHOTT                                                                               LF5   -10.512 1.581336                                                                              -.124312+00                                                                           .118254-01                                                                             .854312-02                                                                            1.384  -14.551                HOYA  FL5   - 10.500                                                                               1.581333                                                                              -.124247+00                                                                           .118332-01                                                                             .915481-02                                                                            1.293  -13.572                SCHOTT                                                                               LF2   -10.439 1.589107                                                                              -.124023+00                                                                           .118811-01                                                                             .819465-02                                                                            1.450  -15.135                OHARA LAF8  -10.387 1.735068                                                                              -.123420+00                                                                           .118826-01                                                                             .865996-02                                                                            1.372  -14.252                OHARA LF2   -10.360 1.589102                                                                              -.123610+00                                                                           .119316-01                                                                             .894944-02                                                                            1.333  -13.812                OHARA BASFO7                                                                               -10.064 1.701415                                                                              -.123199+00                                                                           .122419-01                                                                             .876093-02                                                                            1.397  -14.062                CORN-F                                                                               702411                                                                               -10.014 1.701680                                                                              -.123468+00                                                                           .123294-01                                                                             .109797-01                                                                            1.123  -11.245                SCHOTT                                                                               BASF6 -9.784  1.667435                                                                              -.120949+00 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                                                                            673323                                                                               -4.477  1.672548                                                                              -.155367+00                                                                           .347024-01                                                                             .177009-01                                                                            1.960  -8.777                 HOYA  NBFD9 -4.477  1.756726                                                                              -.157724+00                                                                           .352326-01                                                                             .155215-01                                                                            2.270  -10.162                CHANCE                                                                               850322                                                                               -4.471  1.850071                                                                              -.155534+00                                                                           .347844-01                                                                             .178619-01                                                                            1.947  -8.708                 OHARA LASF2 -4.461  1.850068                                                                              -.155280+00                                                                           .348101-01                                                                             .133600-01                                                                            2.606  -11.623                HOYA  FD19  -4.400  1.666651                                                                              -.151620+00                                                                           .344628-01                                                                             .179466-01                                                                            1.920  -8.448                 SCHOTT                                                                               SF50  -4.391  1.654583                                                                              -.152470+00                                                                           .347218-01                                                                             .160027-01                                                                            2.170  -9.528                 SCHOTT                                                                               SF5   -4.349  1.672547                                                                              -.155622+00                                                                           .357873-01                                                                             .170565-01                                                                            2.098  -9.124                 SCHOTT                                                                               SF62  -4.274  1.681185                                                                              -.156885+00                                                                           .367053-01                                                                             .167365-01                                                                            2.193  -9.374                 HOYA  FD5   -4.266  1.672550                                                                              -.155753+00                                                                           .365086-01                                                                             .179888-01                                                                            2.030  -8.658                 OHARA SF5   -4.255  1.672549                                                                              -.156105+00                                                                           .366839-01                                                                             .177050-01                                                                            2.072  -8.817                 SCHOTT                                                                               LASF32                                                                               -4.205  1.803302                                                                              -.164710+00                                                                           .391719-01                                                                             .491177-02                                                                            7.975  -33.534                SCHOTT                                                                               SFN64 -4.116  1.705682                                                                              -.165302+00                                                                           .401588-01                                                                             .252126-02                                                                            15.928 -65.563                SCHOTT                                                                               BASF54                                                                               -4.095  1.736105                                                                              -.155574+00                                                                           .379897-01                                                                             .114353-01                                                                            3.322  -13.605                CORN-F                                                                               689312                                                                               -4.082  1.688844                                                                              -.160491+00                                                                           .393157-01                                                                             .188028-01                                                                            2.091  -8.535                 SCHOTT                                                                               LASF9 -4.074  1.850066                                                                              -.155146+00                                                                           .380864-01                                                                             .147781-01                                                                            2.577  -10.498                OHARA SF8   -4.050  1.688770                                                                              - .161032+00                                                                          .397613-01                                                                             .175491-01                                                                            2.266  -9.176                 SCHOTT                                                                               SF8   -4.045  1.688772                                                                              -.160524+00                                                                           .396840-01                                                                             .180977-01                                                                            2.193  -8.870                 HOYA  FD8   -4.030  1.688771                                                                              -.160675+00                                                                           .398746-01                                                                             .181835-01                                                                            2.193  -8.836                 CHANCE                                                                               706300                                                                               -3.904  1.705688                                                                              -.166420+00                                                                           .426291-01                                                                             -.125649-02                                                                           -33.927                                                                               132.448                SCHOTT                                                                               SF52  -3.813  1.688358                                                                              -.163202+00                                                                           .428023-01                                                                             .179492-01                                                                            2.385  -9.092                 CORN-F                                                                               836302                                                                               -3.799  1.836206                                                                              -165259+00                                                                            .435053-01                                                                             .274394-02                                                                            15.855 -60.227                CORN-F                                                                               699301                                                                               -3.746  1.698834                                                                              -165739+00                                                                            .442431-01                                                                             .127899-01                                                                            3.459  -12.959                HOYA  FD15  -3.705  1.698777                                                                              -.166066+00                                                                           .448226-01                                                                             .113998-01                                                                            3.932  -14.567                SCHOTT                                                                               SF15  -3.702  1.698783                                                                              -165953+00                                                                            .448271-01                                                                             .116878-01                                                                            3.835  -14.199                OHARA SF15  -3.691  1.698780                                                                              -.165577+00                                                                           .448647-01                                                                             .116099-01                                                                            3.864  -14.262                CHANCE                                                                               699301                                                                               -3.651  1.698778                                                                              -166161+00                                                                            .445060-01                                                                             .144051-01                                                                            3.159  -11.535                OHARA SF1   -3.631  1.717187                                                                              -.169088+00                                                                           .465651-01                                                                             .195078-01                                                                            2.387  -8.668                 SCHOTT                                                                               SF1   -3.624  1.717185                                                                              -.169134+00                                                                           .466728-01                                                                             .194224-01                                                                            2.403  -8.708                 HOYA  FD1   -3.597  1.717184                                                                              -.169250+00                                                                           .470500-01                                                                             .198544-01                                                                            2.370  -8.525                 HOYA  FD20  -3.587  1.720047                                                                              -.170467+00                                                                           .475280-01                                                                             .161718-01                                                                            2.939  -10.541                CHANCE                                                                               717295                                                                               -3.578  1.717189                                                                              -.169016+00                                                                           .472341-01                                                                             .181433-01                                                                            2.603  -9.316                 HOYA  FD18  -3.576  1.721329                                                                              -.170626+00                                                                           .477197-01                                                                             .189655-01                                                                            2.516  -8.997                 SCHOTT                                                                               SF18  -3.561  1.721332                                                                              -.170563+00                                                                           .479035-01                                                                             .193173-01                                                                            2.480  -8.830                 OHARA SF01  -3.533  1.720041                                                                              -.170120+00                                                                           .481532-01                                                                             .196576-01                                                                            2.450  -8.654                 OHARA SF18  -3.531  1.721328                                                                              -.170537+00                                                                           .482918-01                                                                             .197161-01                                                                            2.449  -8.650                 SCHOTT                                                                               TISF1 -3.526  1.673255                                                                              -.172328+00                                                                           .488749-01                                                                             -.942108-02                                                                           -5.188 18.292                 CORN-F                                                                               717295                                                                               -3.512  1.717235                                                                              -.168890+00                                                                           .480868-01                                                                             .128491-01                                                                            3.742  -13.144                SCHOTT                                                                               SF53  -3.416  1.728118                                                                              -.173616+00                                                                           .508275-01                                                                             .165347-01                                                                            3.074  -10.500                HOYA  FDS2  -3.403  1.728117                                                                              -.173734+00                                                                           .510503-01                                                                             .169420-01                                                                            3.013  -10.255                SCHOTT                                                                               LAF9  -3.379  1.794839                                                                              -.175409+00                                                                           .519128-01                                                                             .173355-01                                                                            2.995  -10.118                CORN-F                                                                               728284                                                                               -3.370  1.728119                                                                              -175210+00                                                                            .519861-01                                                                             .122161-01                                                                            4.256  -14.343                HOYA  FD10  -3.347  1.728067                                                                              -.175779+00                                                                           .525203-01                                                                             .102814-01                                                                            5.108  - 17.097               OHARA SF10  -3.331  1.728065                                                                              -.174981+00                                                                           .525319-01                                                                             .137174-01                                                                            3.830  -12.756                CORN-F                                                                               740281                                                                               -3.328  1.739811                                                                              -.176848+00                                                                           .531469-01                                                                             .104692-01                                                                            5.077  -16.892                CHANCE                                                                               728284                                                                               -3.321  1.728063                                                                              -.175405+00                                                                           .528124-01                                                                             .139044-01                                                                            3.798  -12.615                SCHOTT                                                                               SF10  -3.318  1.728065                                                                              -.175217+00                                                                           .528054-01                                                                             .127250-01                                                                            4.150  -13.770                SCHOTT                                                                               SF3   -3.317  1.739812                                                                              -.176532+00                                                                           .532265-01                                                                             .199391-01                                                                            2.669  -8.854                 HOYA  FD3   -3.309  1.739810                                                                              -.176226+00                                                                           .532600-01                                                                             .197209-01                                                                            2.701  -8.936                 OHARA SF3   -3.283  1.739809                                                                              -.175936+00                                                                           .535928-01                                                                             .207575-01                                                                            2.582  -8.476                 SCHOTT                                                                               SF54  -3.279  1.740610                                                                              -.177244+00                                                                           .540529-01                                                                             .188731-01                                                                            2.864  -9.391                 CHANCE                                                                               748278                                                                               -3.259  1.748224                                                                              -.178804+00                                                                           .548708-01                                                                             .210530-01                                                                            2.606  -8.493                 OHARA SFO4  -3.219  1.750644                                                                              -.179544+00                                                                           .557806-01                                                                             .178977-01                                                                            3.117  -10.032                HOYA  FD4   -3.219  1.755001                                                                              -.180410+00                                                                           .560502-01                                                                             .157490-01                                                                            3.559  -11.455                CHANCE                                                                               755276                                                                               -3.206  1.755007                                                                              -.180326+00                                                                           .562432-01                                                                             .196885-01                                                                            2.857  -9.159                 CORN-F                                                                               755276                                                                               -3.197  1.755006                                                                              -.180036+00                                                                           .563099-01                                                                             .123656-01                                                                            4.554  -14.559                SCHOTT                                                                               SF63  -3.193  1.748205                                                                              -.179540+00                                                                           .562238-01                                                                             .191761-01                                                                            2.932  -9.363                 SCHOTT                                                                               SF4   -3.192  1.755004                                                                              -.180351+00                                                                           .564958-01                                                                             .205532-01                                                                            2.749  -8.775                 OHARA SF13  -3.166  1.740577                                                                              -.178843+00                                                                           .564814-01                                                                             .132034-01                                                                            4.278  -13.545                OHARA SF4   -3.154  1.755002                                                                              -.180714+00                                                                           .572879-01                                                                             .196616-01                                                                            2.914  -9.191                 SCHOTT                                                                               SF61  -3.149  1.750645                                                                              -.180774+00                                                                           .574082-01                                                                             .195867-01                                                                            2.931  -9.229                 CORN-F                                                                               762269                                                                               -3.141  1.761598                                                                              -.184460+00                                                                           .587192-01                                                                             .118663-01                                                                            4.948  -15.545                HOYA  FD13  -3.140  1.740582                                                                              -.179728+00                                                                           .572369-01                                                                             .210271-01                                                                            2.722  -8.547                 SCHOTT                                                                               SF13  -3.128  1.740578                                                                              -.179992+00                                                                           .575388-01                                                                             .889124-02                                                                            6.471  -20.244                SCHOTT                                                                               TIF6  -3.054  1.616359                                                                              -.160015+00                                                                           .523949-01                                                                             -.253322-01                                                                           -2.068 6.317                  OHARA SFS5  -3.049  1.761595                                                                              -.183071+00                                                                           .600488-01                                                                             .170045-01                                                                            3.531  -10.766                HOYA  FDS5  -3.046  1.761600                                                                              -.184171+00                                                                           .604674-01                                                                             .131237-01                                                                            4.608  -14.033                SCHOTT                                                                               SF55  -3.040  1.761596                                                                              -.184320+00                                                                           .606329-01                                                                             .189400-01                                                                            3.201  -9.732                 OHARA SFL14 -3.030  1.761615                                                                              -.187228+00                                                                           .617896-01                                                                             -.149891-03                                                                           -412.230                                                                              1249.090               OHARA SFSL3 -2.984  1.784482                                                                              -.188637+ 00                                                                          .632232-01                                                                             .241489-02                                                                            26.181 -78.114                CORN-F                                                                               734265                                                                               -2.969  1.733821                                                                              -.186901+00                                                                           .629605-01                                                                             -.296774-02                                                                           -21.215                                                                               62.978                 CORN-F                                                                               785259                                                                               -2.931  1.784776                                                                              -.191216+00                                                                           .652490-01                                                                             .116685-01                                                                            5.592  -16.387                OHARA SF14  -2.917  1.761613                                                                              -.186558+00                                                                           .639638-01                                                                             .888133-02                                                                            7.202  -21.006                SCHOTT                                                                               SF14  -2.903  1.761614                                                                              -.186734+00                                                                           .643310-01                                                                             .591584-02                                                                            10.874 -31.565                OHARA SFS3  -2.902  1.784486                                                                              -.188982+00                                                                           .651195-01                                                                             .178789-01                                                                            3.642  -10.570                SCHOTT                                                                               SF56  -2.895  1.784485                                                                              -.190232+00                                                                           .657154-01                                                                             .189545-01                                                                            3.467  -10.036                HOYA  FD14  -2.895  1.761617                                                                              -.186961+00                                                                           .645879-01                                                                             .151099-01                                                                            4.275  -12.373                HOYA  FDS3  -2.884  1.784484                                                                              -.190638+00                                                                           .660970-01                                                                             .217325-01                                                                            3.041  -8.772                 OHARA SFL11 -2.879  1.784504                                                                              -.193021+00                                                                           .670424-01                                                                             .456981-03                                                                            146.707                                                                               -422.383               OHARA SFL6  -2.858  1.804954                                                                              -.194939+00                                                                           .682144-01                                                                             049951-02                                                                             14.831 -42.383                CORN-F                                                                               805255                                                                               -2.841  1.804867                                                                              -.194059+00                                                                           .683118-01                                                                             .152341-01                                                                            4.484  -12.738                CHANCE                                                                               805254                                                                               -2.834  1.804952                                                                              -.195132+00                                                                           .688456-01                                                                             .265278-01                                                                            2.595  -7.356                 HOYA  FD11  -2.820  1.784496                                                                              -.191123+00                                                                           .677795-01                                                                             - .118843-01                                                                          -5.703 16.082                 SCHOTT                                                                               SF6   -2.815  1.804954                                                                              -.194617+00                                                                           .691471-01                                                                             .186358-01                                                                            3.710  -10.443                CHANCE                                                                               785258                                                                               -2.810  1.784501                                                                              -.192403+00                                                                           .684671-01                                                                             .128753-01                                                                            5.318  -14.944                HOYA  FD6   -2.803  1.804957                                                                              -.194363+00                                                                           .693396-01                                                                             -.188400-01                                                                           3.680  -10.316                OHARA SF11  -2.796  1.784502                                                                              -.192467+00                                                                           .688348-01                                                                             .584361-02                                                                            11.780 -32.936                SCHOTT                                                                               SF11  -2.787  1.784502                                                                              -.192107+00                                                                           .689216-01                                                                             .605232-02                                                                            11.388 -31.741                OHARA SF6   -2.779  1.804950                                                                              -.194546+00                                                                           .699951-01                                                                             .202121-01                                                                            3.463  -9.625                 SCHOTT                                                                               SF57  -2.563  1.846408                                                                              -.206927+00                                                                           .807402-01                                                                             .162222-01                                                                            4.977  -12.756                HOYA  FDS9  -2.552  1.846403                                                                              -.206914+00                                                                           .810766-01                                                                             .174143-01                                                                            4.656  -11.882                OHARA SFO3  -2.533  1.846404                                                                              -.206695+00                                                                           .816138-01                                                                             .212868-01                                                                            3.781  -9.575                 SCHOTT                                                                               SF58  -2.235  1.917309                                                                              -.228319+00                                                                           .102142+00                                                                             .149349-01                                                                            6.839  -15.288                OHARA SFSO1 -2.208  1.922555                                                                              -.230396+00                                                                           .104355+00                                                                             .113419-01                                                                            9.201  -20.314                OHARA SFS1  -2.128  1.922547                                                                              -.234526+00                                                                           .110212+00                                                                             .744372-02                                                                            14.806 -31.507                SCHOTT                                                                               SF59  -2.109  1.952164                                                                              -.240117+00                                                                           .113849+00                                                                             .162320-02                                                                            70.139 - 147.928              OHARA BK9   2.861   1.493821                                                                              -.803643-01                                                                           -.280896-01                                                                            -.204052-01                                                                           1.377  3.938                  SCHOTT                                                                               FK5   2.865   1.487438                                                                              -.753913-01                                                                           -.263180-01                                                                            -.189047-01                                                                           1.392  3.988                  HOYA  FC5   2.905   1.487438                                                                              -.753744-01                                                                           -.259430-01                                                                            -.188962-01                                                                           1.373  3.989                  CORN-F                                                                               488658                                                                               2.919   1.487447                                                                              -.807348-01                                                                           -.276601-01                                                                            -.194901-01                                                                           1.419  4.142                  HOYA  BSC3  2.952   1.498252                                                                              -.813643-01                                                                           -.275662-01                                                                            -.193049-01                                                                           1.428  4.215                  OHARA FK5   2.960   1.487438                                                                              -.756632-01                                                                           -.255643-01                                                                            -.170814-01                                                                           1.497  4.430                  SCHOTT                                                                               BK10  2.960   1.497766                                                                              -.792124-01                                                                           -.267620-01                                                                            -.192149-01                                                                           1.393  4.122                  SCHOTT                                                                               BK3   2.991   1.498254                                                                              -.814789-01                                                                           -.272409-01                                                                            -.195328-01                                                                           1.395  4.171                  CORN-F                                                                               488704                                                                               2.993   1.498196                                                                              -.752048-01                                                                           -.251274-01                                                                            -.167875-01                                                                           1.497  4.480                  OHARA BK10  3.025   1.497765                                                                              -.792615-01                                                                           -.262056-01                                                                            -.177864-01                                                                           1.473  4.456                  SCHOTT                                                                               PK1   3.075   1.503725                                                                              -.791244-01                                                                           -.257320-01                                                                            -.180104-01                                                                           1.429  4.393                  OHARA BK4   3.126   1.500426                                                                              -.802655-01                                                                           -.256745-01                                                                            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3.185   1.498251                                                                              -.813600-01                                                                           -.255469-01                                                                            -.174240-01                                                                           1.466  4.669                  OHARA PK1   3.188   1.503723                                                                              -.790943-01                                                                           -.248065-01                                                                            -.162424-01                                                                           1.527  4.870                  OHARA PK2   3.242   1.518146                                                                              -.812802-01                                                                           -.250680-01                                                                            -.170427-01                                                                           1.471  4.769                  HOYA  FC1   3.259   1.470636                                                                              -.785432-01                                                                           -.240986-01                                                                            -.159417-01                                                                           1.512  4.927                  OHARA FK3   3.275   1.464446                                                                              -.801760-01                                                                           -.244804-01                                                                            -.161796-01                                                                           1.513  4.955                  SCHOTT                                                                               PK50  3.298   1.520484                                                                              -.757644-01                                                                           -.229694-01                                                                            -.150775-01                                                                           1.523  5.025                  OHARA FK1   3.343   1.470638                                                                              -.784884-01                                                                           -.234806-01                                                                            -.143948-01                                                                           1.631  5.453                  SCHOTT                                                                               FK3   3.377   1.464448                                                                              -.802741-01                                                                           -.237720-01                                                                            -.157250-01                                                                           1.512  5.105                  CORN-F                                                                               465657                                                                               3.379   1.464448                                                                              -.803645-01                                                                           -.237823-01                                                                            -.151281-01                                                                           1.572  5.312                  SCHOTT                                                                               PK2   3.397   1.518151                                                                              -.811341-01                                                                           -.238813-01                                                                            -.160876-01                                                                           1.484  5.043                  HOYA  PC2   3.429   1.518151                                                                              -.809744-01                                                                           -.236126-01                                                                            -.154907-01                                                                           1.524  5.227                  CORN-F                                                                               518651                                                                               3.477   1.518140                                                                              -.810585-01                                                                           -.233110-01                                                                            -.146698-01                                                                           1.589  5.526                  OHARA PKS1  3.486   1.517222                                                                              -.757358-01                                                                           -.217277- 01                                                                           -.135002-01                                                                           1.609  5.610                  HOYA  PCS1  3.493   1.517229                                                                              -.756531-01                                                                           -.216566-01                                                                            -.118306-01                                                                           1.831  6.395                  SCHOTT                                                                               PK3   3.512   1.525358                                                                              -.815369-01                                                                           -.232167-01                                                                            -.151485-01                                                                           1.533  5.383                  OHARA PK3   3.535   1.525355                                                                              -.817364-01                                                                           -.231251-01                                                                            -.153467-01                                                                           1.507  5.326                  HOYA  FC3   3.566   1.464450                                                                              -.802805-01                                                                           -.225129-01                                                                            -.131702-01                                                                           1.709  6.096                  HOYA  PC3   3.588   1.525360                                                                              -.815538-01                                                                           -.227293-01                                                                            -.144528-01                                                                           1.573  5.643                  SCHOTT                                                                               BK7   3.595   1.516740                                                                              -.821247-01                                                                           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                                         1.562  5.660                  OHARA BK12  3.627   1.518679                                                                              -.816145-01                                                                           -.225015-01                                                                            -.143862-01                                                                           1.564  5.673                  CORN-F                                                                               517642                                                                               3.627   1.516744                                                                              -.820742-01                                                                           -.226262-01                                                                            -.138974-01                                                                           1.628  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                                                       517642                                                                               3.704   1.516736                                                                              -.820335-01                                                                           -.221500-01                                                                            -.138473-01                                                                           1.611  5.967                  CHANCE                                                                               641601                                                                               3.706   1.640416                                                                              -.876494-01                                                                           -.236528-01                                                                            -.159457-01                              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                               569631                                                                               4.362   1.568661                                                                              -.831007-01                                                                           -.190507-01                                                                            -.101752-01                                                                           1.872  8.167                  OHARA BK6   4.459   1.531062                                                                              -.839051-01                                                                           -.188158-01                                                                            -.907099-02                                                                           2.074  9.250                  SCHOTT                                                                               K11   4.520   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         TIK1  5.446   1.478630                                                                              -.889197-01                                                                           -.163273- 01                                                                           -.906127-02                                                                           1.802  9.813                  SCHOTT                                                                               LAKN14                                                                               5.459   1.696707                                                                              -.941778-01                                                                           -.172531-01                                                                            -.835559-02                                                                           2.065  11.271                 CHANCE                                                                               603606                                                                               5.467   1.603035                                                                              -.860735-01                                                                           -.157455-01                                                                            -.621115-02                                                                           2.535  13.858                 OHARA SK5   5.472   1.589055                                                                              -.855443-01                                                                           -.156340-01                                                                            -.591099-02                                                                           2.645  14.472                 SCHOTT                                                                               K7    5.485   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                                                                    LAK11 6.233   1.658216                                                                              -.909186-01                                                                           -.145874-01                                                                            -.523021-02                                                                           2.789  17.383                 SCHOTT                                                                               BAK50 6.237   1.567664                                                                              -.897380-01                                                                           -.143880-01                                                                            -.501834-02                                                                           2.867  17.882                 CHANCE                                                                               652585                                                                               6.241   1.651515                                                                              -.889501-01                                                                           -.142526-01                                                                            -.459356-02                                                                           3.103  19.364                 SCHOTT                                                                               LAK9  6.277   1.690909                                                                              -.951366-01                                                                           -.151571-01                                                                            -.622629-02                                                                           2.434  15.280                 OHARA SK12  6.313   1.583056                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       39.614  1.794867                                                                              -.113455+ 00                                                                          -.286402-02                                                                            .359110-02                                                                            -.798  -31.593                OHARA SSK3  40.938  1.614751                                                                              -.100315+00                                                                           -.245043-02                                                                            .383497-02                                                                            -.639  -26.158                SCHOTT                                                                               SSKN8 42.626  1.617629                                                                              -.103142+00                    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OHARA LASF04                                                                               143.537 1.815402                                                                              -.115359+00                                                                           -.803689-03                                                                            .568507-02                                                                            -.141  -20.292                SCHOTT                                                                               BAF54 149.969 1.666619                                                                              -.106286+00                                                                           -.708719-03                                                                            .385235-02                                                                            -.184  -27.590                SCHOTT                                                                               LLF2  171.906 1.540635                                                                              -.108727+00                                                                           -.632476-03                                                                            .125404-02                                                                            -.504  -86.701                CHANCE                                                                               744448                                                                               183.119 1.743877                                                                              -.114507+00                                                                           -.625315-03                                                                            .184080-02                                                                            -.340  -62.205                HOYA  FEL2  184.340 1.540634                                                                              -.108625+00                                                                           -.589262-03                                                                            .142463-02                                                                            -.414  -76.247                CORN-F                                                                               744448                                                                               192.746 1.743879                                                                              -.114440+00                                                                           -.593733-03                                                                            .220180-02                                                                            -.270  -51.976                HOYA  TAFD1 205.243 1.814564                                                                              -.115964+00                                                                           -.565007-03                                                                            .622681-02                                                                            -.091  -18.623                HOYA  NBFD14                                                                               224.677 1.801566                                                                              -.115564+00                                                                           -.514356-03                                                                            .564119-02                                                                            -.091  -20.486                HOYA  TAFD10                                                                               258.348 1.815368                                                                              -.115241+00                                                                           -.446069-03                                                                            .763394-02                                                                            -.058  -15.096                OHARA LLF2  363.647 1.540638                                                                              -.108587+00                                                                           -.298606-03                                                                            .161587-02                                                                            -.185  -67.200                SCHOTT                                                                               LGSK2 452.288 1.585919                                                                              -.837990-01                                                                           -.185278-03                                                                            .107401-01                                                                            -.017  -7.802                 OHARA LAF06 978.209 1.685896                                                                              -.104144+00                                                                           -.106464-03                                                                            .570256-02                                                                            -.019  - 18.263               CORN-F                                                                               655401                                                                               1271.050                                                                               1.655380                                                                              -.127954+00                                                                           -.100597-03                                                                            -.301127-02                                                                           .033   42.492                 __________________________________________________________________________ 

We claim:
 1. An anamorphic optical system comprising a plurality of optical elements, said optical elements being configured and positioned with respect to each other to produce zero chromatic dispersion at more than two discrete wavelengths for light rays passing through said system, said optical elements being made from only two different optical materials.
 2. The optical system of claim 1 wherein said optical elements are prism elements, said prism elements coacting with each other so that light rays entering said system at a specified incident angle emerge from said system with a common total deviation angle relative to said specified incident angle at more than two discrete wavelengths.
 3. An anamorphic optical system comprising a plurality of prism elements, said prism elements being made of only two different optical materials, said prism elements being configured and positioned with respect to each other so that light rays entering said system at a specified incident angle emerge from said system with a substantially identical total deviation angle relative to said specified incident angle at more than two wavelengths.
 4. The anamorphic optical system of claim 3 wherein said plurality of prism elements comprises only two prism elements.
 5. The anamorphic system of claim 3 comprising a first prism element made of an optical material having a refractive index of approximately 1.53 and an Abbe number of approximately 76.57 at a base wavelength of 0.546 micrometers, and a second prism element made of an optical material having a refractive index of approximately 1.65 and an Abbe number of approximately 58.15 at said base wavelength.
 6. The anamorphic system of claim 5 wherein said first prism element has an apex angle of approximately 42.02 degrees, and said second prism element has an apex angle of approximately 25.59 degrees.
 7. The anamorphic system of claim 3 comprising first, second, third, fourth and fifth prism elements, said first, third and fifth prism elements being made of an optical material having a refractive index of approximately 1.53 and an Abbe number of approximately 76.57 at a base wavelength of 0.546 micrometers, said second and fourth prism elements being made of an optical material having a refractive index of approximately 1.65 and an Abbe number of approximately 58.15 at said base wavelength.
 8. The anamorphic system of claim 7 wherein four of said prism elements have apex angles that are substantially equal.
 9. The anamorphic system of claim 8 wherein each of said first, second, third and fourth prism elements has an apex angle of approximately 43.17 degrees, and said fifth prism element has an apex angle of approximately 57.58 degrees.
 10. The anamorphic system of claim 7 wherein said first prism element has an apex angle of approximately 83.20 degrees, said second prism element has an apex angle of approximately 44.96 degrees, said third prism element has an apex angle of approximately 40.86 degrees, said fourth prism element has an apex angle of approximately 46.92 degrees, and said fifth prism element has an apex angle of approximately 37.50 degrees.
 11. A prism system comprising a plurality of prism elements, said prism elements being configured and positioned with respect to each other so that light rays entering said system at a specified incident angle emerge from said system with wavelength-dependent total deviation angles relative to said specified incident angle, chromatic dispersion from a given total deviation angle being zero at five wavelengths for light rays emerging from said systems, said prism elements being made from only two different optical materials.
 12. The prism system of claim 11 wherein said two optical materials are Schott PK51 glass and Hoya LAC7 glass.
 13. The prism system of claim 12 wherein said plurality of prism elements comprises only two prism elements.
 14. The prism system of claim 13 wherein said two prism elements are configured and positioned with respect to each other so as to produce a common total deviation angle of six degrees with respect to said incident angle for light rays passing through said system at said five discrete wavelengths.
 15. The prism system of claim 12 wherein said plurality of prism elements comprises five prism elements.
 16. The prism system claim 15 wherein said five prism elements are configured and positioned with respect to each other so as to produce a common total deviation angle of thirty degrees with respect to said incident angle for light rays passing through said system at said five discrete wavelengths. 